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Mazda RX-7

The Mazda RX-7 is a front-engine, rear-wheel-drive, rotary engine-powered sports car that was manufactured and marketed by Mazda from 1978 until 2002 across three generations, all of which made use of a compact, lightweight Wankel rotary engine.

The first-generation RX-7, sometimes referred to as the SA (early) and FB (late), is a two-seater two-door hatchback coupé. It featured a 12A carbureted rotary engine as well as the option for a 13B rotary engine with electronic fuel injection in later years.

The second-generation RX-7, sometimes referred to as the FC, was offered as a two-seater coupé with a 2+2 option available in some markets, as well as in a convertible body style. This was powered by the 13B rotary engine, offered in naturally aspirated or turbocharged forms.

The third-generation RX-7, sometimes referred to as the FD, was offered a 2+2-seater coupé with a limited run of a two-seater option. Some markets were only available as a two-seater. It featured a sequentially-turbocharged 13B REW engine.

More than 800,000 RX-7s were manufactured over its lifetime.

Series 1 (1978–1980)
Series 1 (produced from 1978 until 1980) is commonly referred to as the "SA22C" from the first alphanumeric of the vehicle identification number. Mazda's internal project number for what was to become the RX-7 was X605. In Japan, it was introduced in March 1978, replacing the Savanna RX-3, and joined Mazda's only other remaining rotary engine-powered products, called the Cosmo which was a two-door luxury coupé, and the Luce luxury sedan.

The lead designer at Mazda was Matasaburo Maeda (前田 又三郎), whose son, 前田育男, would go on to design the Mazda2 and the RX-7's successor, the RX-8. The transition of the Savanna to a sports car appearance reflected products from other Japanese manufacturers. The advantage the RX-7 had was its minimal size and weight, and the compact rotary engine installed behind the front axle helped balance the front and rear weight distribution, which provided a low center of gravity.

In May 1980, Mazda introduced a limited production run of special North American models known as the Leathersport Models. This package was essentially an uprated GS model with added LS badges on each B-pillar, special stripes on the exterior, and LS-only gold anodized wheels (with polished outer face and wheel rim). All LS editions came equipped with special LS-only full brown leather upholstery, leather-wrapped steering wheel, leather-wrapped shift knob, removable sunroof, LS-specific four-speaker AM/FM stereo radio with power antenna (though listed as a six-speaker stereo, as the two rear dual voice coil speakers were counted as four speakers in total), remote power door side mirrors, and other standard GS equipment. Two primary options were also available; a three-speed JATCO 3N71B automatic transmission and air conditioning. Other GS options such as cassette tape deck, splash guards, padded center console arm rest and others could be added by the dealer. The LS model was only ever available in three different exterior colors: Aurora White, Brilliant Black, and Solar Gold. No official production records are known to exist or to have been released. This series of RX-7 had exposed steel bumpers and a high-mounted indentation-located rear license plate, called by Werner Buhler of Road & Track magazine a "Baroque depression."

Series 2 (1981–1983)

The Series 2, referred to as the FB (produced from 1981 to 1983), had integrated plastic-covered bumpers, wide black rubber body side moldings, wraparound taillights and updated engine control components. While marginally longer overall, the new model was 135 lb lighter in federalized trim. The four-speed manual option was dropped for 1981 as well, while the gas tank grew larger and the dashboard was redesigned, including a shorter gear stick mounted closer to the driver. In 1983, the 130 mph speedometer returned for the RX-7. The GSL package provided optional four-wheel disc brakes, front ventilated (Australian model) and clutch-type rear limited-slip differential (LSD). This revision of the SA22 was known in North America as the "FB" after the US Department of Transportation mandated 17 digit Vehicle Identification Number changeover. For various other markets worldwide, the 1981–1985 RX-7 retained the 'SA22C' VIN prefix. In the UK, the 1978–1980 series 1 cars carried the SA code on the VIN but all later cars (1981–1983 series 2 and 1984–1985 series 3) carried the FB code and these first-generation RX-7s are known as the "FB" only in Northern America.

In Japan, a very well appointed version similar to the export market GSL arrived late in 1982, called the SE-Limited. This model received two-tone paint, alloy wheels shaped like the Wankel rotor, all-wheel disc brakes, limited-slip differential, and a full leather interior. It also had the latest iteration of the 12A rotary engine, the RE-6PI with variable induction port system and 140 PS.

In Europe, the FB was mainly noticed for having received a power increase from the 105 PS of the SA22; the 1981 RX-7 now had 115 PS on tap. European market cars also received four-wheel disc brakes as standard.

Series 3 (1984–1985)

The Series 3 (produced 1984–1985) featured an updated lower front fascia. North American models received a different instrument cluster. GSL package was continued into this series, but Mazda introduced the GSL-SE sub-model. The GSL-SE had a fuel injected 1308 cc 13B RE-EGI engine rated at 135 hp and 180 Nm. GSL-SE models had much the same options as the GSL (clutch-type rear LSD and rear disc brakes), but the brake rotors were larger, allowing Mazda to use the more common lug nuts (versus bolts), and a new bolt pattern of 4x114.3mm (4x4.5"). Also, they had upgraded suspension with stiffer springs and shocks. The external oil cooler was reintroduced, after being dropped in the 1983 model-year for the controversial "beehive" water-oil heat exchanger.

The 1984 RX-7 GSL has an estimated 29 MPG (8.11 litres/100 km) highway/19 MPG (12.37 L/100 km) city. According to Mazda, its rotary engine, licensed by NSU-Wankel allowed the RX-7 GSL to accelerate from 0 to 80 km/h (50  mph) in 6.3 seconds.

In 1985, Mazda introduced the RX-7 Finale in Australia. This was the last of the series and brought out in limited numbers. The Finale featured power options and a brass plaque mentioning the number the car was as well as "Last of a legend" on the plaque. The finale had special stickers and a blacked out section between the window & rear hatch.

The handling and acceleration of the car were noted to be of a high caliber for its day. The RX-7 had "live axle" 4-link rear suspension with Watt's linkage, a 50:50 front and rear weight distribution, and weighed under 1100 kg. It was the lightest generation of the RX-7 ever produced. 12A-powered models accelerated from 0–97 km/h (60 mph) in 9.2 seconds, and turned 0.779 g (7.64 m/s²) laterally on a skid pad. The 1146 cc 12A engine was rated at 100 hp at 6,000 rpm in North American models, allowing the car to reach speeds of over 120 mph. Because of the smoothness inherent in the Wankel rotary engine, little vibration or harshness was experienced at high engine speeds, so a buzzer was fitted to the tachometer to warn the driver when the 7,000 rpm redline was approaching.

The 12A engine has a long thin shaped combustion chamber, having a large surface area in relation to its volume. Therefore, combustion is cool, giving few oxides of nitrogen. However, the combustion is also incomplete, so there are large amounts of partly burned hydrocarbons and carbon monoxide. The exhaust is hot enough for combustion of these to continue into the exhaust. An engine-driven pump supplies air into the exhaust to complete the burn of these chemicals. This is done in the "thermal reactor" chamber where the exhaust manifold would normally be on a conventional engine. Under certain conditions, the pump injects air into the thermal reactor and at other times air is pumped through injectors into the exhaust ports. This fresh air is needed for more efficient and cleaner-burning of the air/fuel mixture.

Options and models varied from country to country. The gauge layout and interior styling in the Series 3 was only changed for the North American models. Additionally, North America was the only market to have offered the first generation of the RX-7 with the fuel-injected 13B, model GSL-SE. Sales of the first-generation RX-7 were strong, with a total of 474,565 cars produced; 377,878 (nearly eighty percent) were sold in the United States alone.

RX-7 Turbo

Following the introduction of the first turbocharged rotary engine in the Luce/Cosmo, a similar, also fuel injected and non-intercooled 12A turbocharged engine was made available for the top-end model of the Series 3 RX-7 in Japan. It was introduced in September 1983. The engine was rated at 165 PS (JIS) at 6,500 rpm. While the peak power figures were only slightly higher than those of the engine used in the Luce/Cosmo, the new "Impact Turbo" was developed specifically to deal with the different exhaust gas characteristics of a rotary engine. Both rotor vanes of the turbine were remodeled and made smaller, and the turbine had a twenty percent higher speed than a turbo intended for a conventional engine. The Savanna Turbo was short-lived, as the next generation of the RX-7 was about to be introduced.

Second generation (FC)

The second-generation RX-7 (sometimes referred to as "FC", VIN begins JM1FC3 or JMZFC1), still known as the Mazda Savanna RX-7 in Japan, featured a complete restyling much like similar sports cars of the era such as the Nissan 300ZX. Mazda's development team, led by Chief Project Engineer Akio Uchiyama (内山 昭朗), chose to focus on the American market when designing the FC, where the majority of first-generation of the RX-7 models had been sold. The team drew inspiration from successful sports cars that were popular at the time, such as studying the suspension design of the Porsche 928.

While the first-generation RX-7 was a purer sports car, the second-generation RX-7 tended toward the softer sport-tourer trends of its day, sharing some similarities with the HB series Cosmo. Handling was much improved, with less of the oversteer tendencies of the previous version. The rear end design was vastly improved from the SA22's live rear axle to a more modern, Independent Rear Suspension (rear axle). Steering was more precise, with rack and pinion steering replacing the old recirculating-ball steering of the SA22. Disc brakes also became standard, with some models (S4: Sport, GXL, GTU, Turbo II, Convertible; S5: GXL, GTUˢ, Turbo, Convertible) offering four-piston front brakes. The rear seats were optional in some models of the FC RX-7, but are not commonly found in the American Market. Mazda also introduced Dynamic Tracking Suspension System (DTSS) in the FC. The revised independent rear suspension incorporated special toe control hubs which were capable of introducing a limited degree of passive rear steering under cornering loads. The DTSS worked by allowing a slight amount of toe-out under normal driving conditions but induced slight toe-in under heavier cornering loads at around 0.5g or more; toe-out in the rear allows for a more responsive rotation of the rear, but toe-in allowed for a more stable rear under heavier cornering. Another new feature was the Auto Adjusting Suspension (AAS). The system changed damping characteristics according to the road and driving conditions. The system compensated for camber changes and provided anti-dive and anti-squat effects.

In Japan, a limited edition of the second-generation RX-7 called Infini was available with production limited to only 600 cars for each year. Some special noted features for all Infini series are: infini logo on the rear, upgraded suspension, upgraded ECU, higher power output of the engine, reduced weight, 15-inch BBS aluminum alloy wheels, Infini logo steering wheel, aero bumper kits, bronze-colored window glass, floor bar on the passenger side, aluminum bonnet with scoop, flare, and holder. The car was thought as the pinnacle of the RX-7 series (until the introduction of the third generation RX-7). The Infini IV came with other special items such as black bucket seats, 16-inch BBS wheels, Knee pads, and all the other items mentioned before. There are differing years for the Infini, which denoted the series. Series I was introduced in 1987, Series II was introduced in 1988, Series III was introduced in 1989, and Series IV was introduced in 1990. Series I and II came in White or Black exterior colour, Series III came in Shade Green only, and Series IV came in Shade Green or Noble Green exterior colours. There are only minor differences between the Series models, the biggest change which was from the Series I and II being an S4 and the Series III and IV being an S5.

The Turbo II model uses a turbocharger with a twin-scroll design. The smaller primary chamber is engineered to cancel the turbo lag at low engine speeds. At higher revolutions, the secondary chamber is opened, pumping out 33 percent more power than the naturally aspirated counterpart. The Turbo II also has an air-to-air intercooler which has a dedicated intake on the hood. The intake is slightly offset toward the left side of the hood. In the Japanese market, only the turbocharged engine was available; the naturally-aspirated version was only available for select export markets. This can be attributed to insurance companies in many Western nations penalizing turbocharged cars (thus restricting potential sales). The Japanese market car produces 185 PS in the original version; this engine was upgraded to 205 PS in April 1989 as part of the Series 5 facelift. The limited edition, two-seater Infini model received a 215 PS version beginning in June 1990, thanks to an upgraded exhaust system and high-octane fuel.

Australian Motors Mazda introduced a limited run of 250 'Sports' model Series 4 RX-7s; each with no power steering, power windows or rear wiper as an attempt to reduce the weight of the car.

Convertible
Mazda introduced a convertible version of the RX-7, the FC3C, in 1988 with a naturally aspirated engine—introduced to the US market with ads featuring actor James Garner, at the time featured in many Mazda television advertisements. Only about 22,000 convertibles were built.

The convertible featured a removable rigid section over the passengers and a folding fabric rear section with heated rear glass window. Power operated, lowering the top required unlatching two header catches, power lowering the top, exiting the car (or reaching over to the right-side latch), and folding down the rigid section manually. Mazda introduced with the convertible the first integral windblocker, a rigid panel that folded up from behind the passenger seats to block unwanted drafts from reaching the passengers—thereby extending the driving season for the car with the top retracted. The convertible also featured optional headrest-mounted audio speakers and a folding leather snap-fastened tonneau cover. The convertible assembly was precisely engineered and manufactured, and dropped into the ready body assembly as a complete unit—a first in convertible production.

Production ceased in 1991 after Mazda marketed a limited run of 500 examples for 1992 for the domestic market only. In markets outside the US, only the turbocharged version of the convertible was available.

North America
The Series 4 (produced for the 1986 through the 1988 model years) was available with a naturally aspirated, fuel injected 13B-VDEI producing 146 hp in North American spec. An optional turbocharged model, known as the Turbo II in the American market, was rated at 182 hp and 183 lbft of torque at 3,500 rpm. The turbo model was introduced at the Chicago Auto Show in February 1986, with a target of 20 percent of overall RX-7 sales.

The Series 5 (1989–1992) featured updated styling and better engine management, as well as lighter rotors and a higher compression ratio 9.7:1 for the naturally aspirated model, and 9.0:1 for the turbo model. The naturally aspirated Series 5's 13B-DEI engine was rated at 160 hp, while the Series 5 Turbo was rated at 200 hp at 6,500 rpm and 195 lbft of torque at 3,500 rpm.

Mazda sold 86,000 RX-7s in the US alone in 1986, its first model year, with sales peaking in 1988.

1988 10th Anniversary RX-7
Mazda introduced the 10th Anniversary RX-7 in 1988 as a limited production model based on the RX-7 Turbo II. Production was limited to 1,500 units. The 10th Anniversary RX-7 features a Crystal White monochromatic paint scheme with matching white body side moldings, tail light housings, mirrors and 16-inch alloy seven-spoke wheels. There were two "series" of 10th Anniversary models, with essentially a VIN-split running production change between the two. The most notable difference between the series can be found on the exterior- the earlier "Series I" cars had a black "Mazda" logo decal on the front bumper cover, whereas most if not all "Series II" cars did not have the decal. Series II cars also received the lower seat cushion height/tilt feature that Series I cars lacked. Another distinctive exterior feature is the bright gold rotor-shaped 10th Anniversary Edition badge on the front fenders (yellow-gold on the Series II cars). A distinctive 10th Anniversary package feature is the all black leather interior (code D7), which included not just the seats, but the door panel inserts as well and a leather-wrapped MOMO steering wheel (with 10th Anniversary Edition embossed horn button) and MOMO leather-wrapped shift knob with integrated boot. All exterior glass is bronze tinted (specific in North America to only the 10th Anniversary), and the windshield was equipped with the embedded secondary antenna also found on some other select models with the upgraded stereo packages. Other 10th Anniversary Edition specific items were headlight washers (the only RX-7 in the US market that got this feature), glass breakage detectors added to the factory alarm system, 10th Anniversary Edition logoed floor mats, 10th Anniversary Edition embroidered front hood protector and accompanying front end mask (or "bra"), and an aluminum under pan.

GTUˢ (1989–1990)
In 1989, with the introduction of the facelifted FC RX-7, and to commemorate the RX-7s 8 straight wins in the IMSA GTU class from 1980 to 1987, Mazda introduced a limited model labeled the GTUˢ. Starting with the lightweight base model GTU, the GTUˢ added items found on the Turbo model such as four-piston front brakes, ventilated rear brake rotors, vehicle speed-sensing power steering, one-piece front chin spoiler, cloth-covered Turbo model seats, leather-wrapped steering wheel, 16-inch wheels, 205/55VR tires, and a GTUˢ-only 4.300 viscous-type limited-slip differential (all other FC LSD's were 4.100). This allowed quicker acceleration from the naturally aspirated 13B rotary engine. Although it has been rumored that Mazda built between 100 and 1100 GTUˢ' between 1989 and 1990, there is no official documentation from Mazda on how many have been built. The most reliable way to determine a GTUˢ is by taking the VIN to a Mazda dealer and they can look to see if the internal coding for the car comes up as a GS model designation.

Third generation (FD)
The third-generation RX7, (sometimes referred to as FD, chassis code FD3S for Japan and JM1FD for the North America), featured an updated body design. Announced in October 1991, production began later that month before going on sale in December in the domestic Japanese market. Left-Hand-Drive export production began shortly after. The 13B-REW engine was the first-ever mass-produced sequential twin-turbocharger system to be exported from Japan, boosting power to 255 PS in 1992 and finally to 280 PS by the time production ended in Japan in 2002.

For the third-generation RX-7, Mazda organized an internal design competition between its four design studios in Hiroshima, Yokohama, Irvine, and Europe. The winning design came from their Design Center in Irvine and was designed by Taiwanese automotive artist Wu-huang Chin (秦無荒), who also worked on the Mazda MX-5 Miata, with help from Tom Matano. The interior design, though, originated from the Hiroshima design proposal. Mazda's chief designer Yoichi Sato (佐藤 洋一) then helped take the concept design to its final production form.

In Japan, sales were affected by this series' non-compliance with Japanese dimension regulations and Japanese buyers paid annual taxes for the car's non-compliant width. As the RX7 was now considered an upper-level luxury sports car due to the increased width dimensions, Mazda also offered two smaller offerings, the Eunos Roadster, and the Eunos Presso hatchback.

The sequential twin-turbocharging system, introduced in 1992, was extremely complex and was developed with the aid of Hitachi. It was previously used on the exclusive-to-Japan Cosmo JC Series. The system used two turbochargers, one to provide 10 psi of boost from 1,800 rpm. The second turbocharger activated in the upper half of the rpm range, during full-throttle acceleration – at 4,000 rpm to maintain 10 psi until redline. The changeover process occurred at 4,500 rpm, with a momentary dip in pressure to 8 psi, and provided semi-linear acceleration from a wide torque curve throughout the entire rev range under normal operation.

Under high-speed-driving conditions, the changeover process produced a significant increase in power output and forced technical drivers to adjust their driving style to anticipate and mitigate any over-steer during cornering. The standard turbo control system used 4 control solenoids, 4 actuators, both a vacuum and pressure chamber, and several feet of preformed vacuum/pressure hoses, all of which were prone to failure in part due to complexity and the inherent high temperatures of the rotary engine.

Information about various trims and models is listed as follows:

Series 6 (1992–1995)
Series 6 was exported throughout the world and sold in larger numbers than in later years. In Japan, Mazda sold the RX-7 through its ɛ̃fini brand as the ɛ̃fini RX-7. Models in Japan included the Type S, the base model, Type R, the lightweight sports model, Type RZ, Type RB, A-spec and the Touring X, which came with a four-speed automatic transmission. The RX-7 was sold in 1993–1995 in the U.S. and Canada. The Series 6 was rated at 255 PS and 294 Nm.

North American market
At launch, three option packages were offered; the unlabeled base model, the Touring and the R1 (renamed R2 in 1994). All cars were only available as a two-seater unlike the previous generation which was a 2+2 configuration in North America. All cars were equipped with the same sequential twin-turbo 13B REW engine. A five-speed manual transmission was standard and a 4-speed automatic was available on the base model and Touring package. A driver-side airbag and anti-lock brakes are standard as well. The Touring package included a glass moonroof, fog lights, leather seats, a rear window wiper and a Bose Acoustic Wave music system with CD player. The R1 (R2 in 1994–95) model featured upgraded springs, Bilstein shocks, an additional engine oil cooler, an aerodynamics package comprising a front lip and rear wing, suede seats and Pirelli Z-rated tires. Cruise control was deleted on the R1. The R2 differed from the R1 in that it had slightly softer suspension.

In 1994, the interior received a small update to include a passenger-side air bag, and a PEG (popular equipment group) package was offered. The PEG package featured leather seats, a rear cargo cover and a power steel sunroof. It did not include the fog lights or Bose stereo of the touring package. An automatic transmission was not available with the PEG.

In 1995, the Touring package was replaced by the PEP (popular equipment package). The PEP included a rear wing, leather seats, sunroof and fog lights, but didn't have the Bose Stereo nor the rear window wiper. An estimated 500 RX-7s were produced for the 1995 model year. This would be the final year of RX-7 production for North America.

Australian market
A special high-performance version of the RX-7 was introduced in Australia in 1995, named the RX-7 SP. This model was developed to achieve homologation for racing in the Australian GT Production Car Series and the Eastern Creek 12 Hour production car race. An initial run of 25 cars were made, and later an extra 10 were built by Mazda due to demand. The RX-7 SP was rated at 277 PS and 357 Nm of torque, a substantial increase over the standard model. Other changes included a race-developed carbon fiber nose cone and rear spoiler, a carbon fibre 120 l fuel tank (as opposed to the 76 l tank in the standard car), a 4.3:1 final drive ratio, 17-inch wheels, larger brake rotors and calipers. A "three times more efficient" intercooler, a new exhaust, and a modified ECU were also included. Weight was reduced significantly with the aid of further carbon fibre usage including lightweight vented bonnet and Recaro seats to reduce weight to 1218 kg (from 1250 kg) making this model road-going race car that matched the performance of the rival Porsche Carrera RS Club Sport for the final year Mazda officially entered. The formula paid off when the RX-7 SP won the 1995 Eastern Creek 12 Hour, giving Mazda the winning 12-hour trophy for a fourth straight year. The winning car also gained a podium finish at the international tarmac rally Targa Tasmania months later. A later special version, the Bathurst R, was introduced in 2001 to commemorate this victory in Japan only. It was based on the RX-7 Type R and 500 were built in total, featuring adjustable dampers, a carbon fibre shift knob, carbon fibre interior trim, special fog lamps and a different parking brake lever.

European market
In Europe, only 1,152 examples of the FD were sold through the official Mazda network, due to a high price and a fairly short time span. Only one model was available and it included twin oil-coolers, electric sunroof, cruise control and the rear storage bins in place of the back seats. It also has the stiffer suspension and strut braces from the R models. Germany topped the sales with 446 cars, while UK is second at 210 and Greece third with 168 (thanks to that country's tax structure which favored the rotary engine). The European models also received the 1994 interior facelift, with a passenger air bag. Sales in most of Europe ended after 1995 as it would have been too expensive to reengineer the car to meet the new Euro 2 emissions regulations.

United Kingdom
In the United Kingdom, for 1992, customers were offered only one version of the FD, which was based on a combination of the US touring and the base model. For the following year, in a bid to speed up sales, Mazda reduced the price of the RX-7 to £25,000, down from £32,000, and refunded the difference to those who bought the car before that was announced. From 1992 to 1995, only 210 FD RX-7s were officially sold in the UK. The FD continued to be imported to the UK until 1996. In 1998, for a car that had suffered from slow sales when it was officially sold, with a surge of interest and the benefit of a newly introduced SVA scheme, the FD would become so popular that there were more parallel and grey imported models brought into the country than Mazda UK had ever imported.

Series 7 (1996–1998)
Series 7 included minor changes to the car. Updates included a simplified vacuum routing manifold and a 16-bit ECU which combined with an improved intake system netted an extra 10 PS. This additional horsepower was only available on manual transmission cars as the increase in power was only seen above 7,000 rpm, which was the redline for automatic transmission equipped cars. The rear spoiler and tail lights were also redesigned. The Type RZ model was now equipped with larger brake rotors as well as 17-inch BBS wheels. In Japan, the Series 7 RX-7 was marketed under the Mazda and ɛ̃fini brand name.

Series 7 RX-7s were also sold in Australia, New Zealand and the UK. Series 7 models were produced only with right-hand drive (RHD), and were only exported to RHD markets.

Series 8 (1998–2002)
Series 8 was the final series, and was only available in the Japanese market. More efficient turbochargers were available on certain models, while improved intercooling and radiator cooling was made possible by a redesigned front fascia with larger openings. The seats, steering wheel, and instrument cluster were all changed. The rear spoiler was modified and gained adjustability on certain models. Three horsepower levels are available: 255 PS for automatic transmission equipped cars, 265 PSfor the Type RB, and 280 PS available on the top-of-the-line sporting models.

The high-end "Type RS" came equipped with Bilstein suspension and 17-inch wheels as standard equipment, and reduced weight to 1280 kg. Power was increased with the addition of a less restrictive muffler and more efficient turbochargers which featured abradable compressor seals, 280 PS at 6,500 rpm and 314 Nm of torque at 5000 rpm as per the maximum Japanese limit. The Type RS had a brake upgrade by increasing rotor diameter front and rear to 314 mm and front rotor thickness from 22 mm to 32 mm. The Type RS version also sported a 4.30 final drive ratio, providing a significant reduction in its 0–100 km/h time. The gearbox was also modified, 5th gear was made longer to reduce cruising rpm and improve fuel efficiency. The very limited edition Type RZ version included all the features of the Type RS, but at a lighter weight, at 1270 kg. It also featured gun-metal colored BBS wheels and a red racing themed interior. An improved ABS system worked by braking differently on each wheel, allowing the car better turning during braking. The effective result made for safer driving for the average buyer.

Easily the most collectible of all the RX-7s was the last model limited to 1,500 units. Dubbed the "Spirit R", they combined all the extra features Mazda had used on previous limited-run specials with new exclusive features like cross-drilled brake rotors. Sticker prices when new were 3,998,000 yen for Type-A and B and 3,398,000 yen for Type-C. Mazda's press release said "The Type-A Spirit R model is the ultimate RX-7, boasting the most outstanding driving performance in its history."

There are three models of "Spirit R": the "Type A", "Type B", and "Type C". The "Type A" is a two-seater with a five-speed manual transmission. It features lightweight red trim Recaro front seats as seen in the earlier RZ models. The "Type B" shares all features of the "Type A" but with a 2+2 seat configuration. The "Type C" is also a 2+2, but has a four-speed automatic transmission. Of the 1,504 Spirit R's made, 1,044 were Type A, 420 Type B and 40 Type C. An exclusive Spirit R paint color, Titanium Grey, adorned 719 of the 1,504 cars produced.

In Japan the FD3S production span is categorized into six models: #1 from 1991/12, #2 from 1993/08, #3 from 1995/03, #4 from 1996/01, #5 from 1998/12 and #6 from 2000/10. The model number (1 to 6) shows as the first digit of the six-characters-long JDM VIN; e.g., in VIN FD3S-ABCDEF, the A is the model number. A total of nine limited editions (type RZ in 1992/10 (300 cars), RZ 1993/10 (150), R-II Bathurst 1994/09 (350), R Bathurst X 1995/07 (777), RB Bathurst X 1997/01 (700), RS-R 1997/10 (500), RZ 2000/10 (325), R Bathurst R 2001/08 (650), Spirit R 2002/04 (1504)) and 2 special editions (Bathurst R 1995/02, R Bathurst 2001/12 (2174)) were produced.

Reviews and awards
The RX-7 made Car and Driver magazine's Ten Best list five times (1983, 1987, 1993–1995).

In 2004, Sports Car International named the Series 3 car seventh on their list of Top Sports Cars of the 1970s. In 1983, the RX-7 would appear on Car and Driver magazine's Ten Best list for the first time in 20 years.

Though about 800 lb heavier and more insulated than its predecessor, the FC continued to win accolades from the press. The RX-7 FC was Motor Trend's Import Car of the Year for 1986, and the Turbo II was on Car and Driver magazine's 10 Best list for a second time in 1987.

The FD RX-7 was Motor Trend's Import Car of the Year for 1993. When Playboy first reviewed the FD RX-7 in 1993, they tested it in the same issue as the [then] new Dodge Viper. In that issue, Playboy declared the RX-7 to be the better of the two cars. It went on to win Playboy's Car of the Year for 1993.

The FD RX-7 also made Car and Driver's Ten Best list for 1993 through 1995 – all three years in which it was sold in the United States. In June 2007, Road & Track proclaimed: "The ace in Mazda's sleeve is the RX-7, a car once touted as the purest, most exhilarating sports car in the world." After its introduction in 1991, it won the Automotive Researchers' and Journalists' Conference Car of the Year award in Japan.

Handling in the FD RX-7 was regarded as world-class, and it is still regarded as being one of the finest-handling and best-balanced cars of all time.

Motorsport
Racing versions of the first-generation RX-7 were entered at the prestigious 24 hours of Le Mans endurance race. The first outing for the car, equipped with a 13B engine, failed by less than one second to qualify in 1979. The next year, a 12A-equipped RX-7 not only qualified, it placed 21st overall. That same car did not finish in 1981, along with two more 13B cars. Those two cars were back for 1982, with one 14th-place finish and another DNF. The RX-7 Le Mans effort was replaced by the 717C prototype for 1983.

Mazda began racing RX-7s in the IMSA GTU series in 1979. In its first year, RX-7s placed first and second at the 24 Hours of Daytona, and claimed the GTU series championship. The car continued winning, claiming the GTU championship seven years in a row. The RX-7 took the GTO championship ten years in a row from 1982. In addition to this, a GTX version was developed, named the Mazda RX-7 GTP; this was unsuccessful, and the GTP version of the car was also unsuccessful. The RX-7 has won more IMSA races than any other car model. In the USA SCCA competition RX-7s were raced with great success by Don Kearney in the NE Division and John Finger in the SE Division. Pettit Racing won the GT2 Road Racing Championship in 1998. The car was a '93 Mazda RX-7 street car with only bolt-on accessories. At season end Pettit had 140 points—63 points more than the second place team. This same car finished the Daytona Rolex 24-hour race four times.

The RX-7 also fared well at the Spa 24 Hours race. Three Savanna/RX-7s were entered in 1981 by Tom Walkinshaw Racing. After hours of battling with several BMW 530is and Ford Capris, the RX-7 driven by Pierre Dieudonné and Tom Walkinshaw won the event. Mazda had turned the tables on BMW, who had beaten Mazda's Familia Rotary to the podium eleven years earlier at the same event. TWR's prepared RX-7s also won the British Touring Car Championship in 1980 and 1981, driven by Win Percy.

Canadian-born Australian touring car driver Allan Moffat was instrumental in bringing Mazda into the Australian touring car scene which ran to Group C regulations unique to Australia. Over a four-year span beginning in 1981, Moffat took the Mazda RX-7 to victory in the 1983 Australian Touring Car Championship, as well as a trio of Bathurst 1000 podiums, in 1981 (3rd with Derek Bell), 1983 (second with Yoshimi Katayama) and 1984 (third with former motorcycle champion Gregg Hansford). Privateer racer Peter McLeod drove his RX-7 to win the 1983 Australian Endurance Championship, while Moffat won the Endurance title in 1982 and 1984. Australia's adoption of international Group A regulations, combined with Mazda's reluctance to homologate a Group A RX-7 (meaning that a base number of 5,000 had to be built, plus another 500 "evolution" models), ended Mazda's active participation in Australian touring car racing at the end of the 1984 season. Plans had been in place to replace the RX-7 with a Mazda 929, but testing by Allan Moffat in late 1984 had indicated that the car would be uncompetitive and Mazda abandoned plans to race in Group A.

The RX-7 even made an appearance in the World Rally Championship. The car finished 11th on its debut at the RAC Rally in Wales in 1981. Group B received much of the focus for the first part of the 1980s, but Mazda did manage to place third at the 1985 Acropolis Rally, and when the Group B was folded, its Group A-based replacement, the 323 4WD claimed the victory at Swedish Rally in both 1987 and 1989.

IMSA Bridgestone Supercar Series
The third-generation Mazda RX-7 entered its first professional race in the world on February 23, 1992, at the Miami Grand Prix. The cars made it to the podium many times and won the IMSA Supercar race at Sebring in 1994. Peter Farrell Motorsport also fielded RX-7's in the IMSA Firestone Firehawk Endurance Series dominating many races and finishing runner up in the overall championship two years in a row.

Revivals
Mazda has made several references to a revival of the RX-7 in various forms over the years since the RX-8 was discontinued. In November 2012, MX-5 program manager Nobuhiro Yamamoto indicated that Mazda was working on a 16X based RX-7, with 300 horsepower.

In October 2015, Mazda unveiled the RX-Vision concept car at the Tokyo Motor Show, powered by a new rotary engine and featured design cues reminiscent of the third generation RX-7. A production-ready concept could have followed suit by 2017, marking 50 years since the revealing of Mazda's first rotary-powered sports car, the Cosmo.

In October 2023, Mazda unveiled the Iconic SP concept car at the Japan Mobility Show, powered by a two-rotor rotary engine and also features design cues reminiscent of the third generation RX-7, most notably its pop-up headlights.

title: Mazda RX-7


text:
Effective results in number theory

For historical reasons and in order to have application to the solution of Diophantine equations, results in number theory have been scrutinised more than in other branches of mathematics to see if their content is effectively computable. Where it is asserted that some list of integers is finite, the question is whether in principle the list could be printed out after a machine computation.

Littlewood's result
An early example of an ineffective result was J. E. Littlewood's theorem of 1914, that in the prime number theorem the differences of both ψ(x) and π(x) with their asymptotic estimates change sign infinitely often. In 1933 Stanley Skewes obtained an effective upper bound for the first sign change, now known as Skewes' number.

In more detail, writing for a numerical sequence f (n), an effective result about its changing sign infinitely often would be a theorem including, for every value of N, a value M > N such that f (N) and f (M) have different signs, and such that M could be computed with specified resources. In practical terms, M would be computed by taking values of n from N onwards, and the question is 'how far must you go?' A special case is to find the first sign change. The interest of the question was that the numerical evidence known showed no change of sign: Littlewood's result guaranteed that this evidence was just a small number effect, but 'small' here included values of n up to a billion.

The requirement of computability is reflected in and contrasts with the approach used in the analytic number theory to prove the results. It for example brings into question any use of Landau notation and its implied constants: are assertions pure existence theorems for such constants, or can one recover a version in which 1000 (say) takes the place of the implied constant? In other words, if it were known that there was M > N with a change of sign and such that

 * M = O(G(N))

for some explicit function G, say built up from powers, logarithms and exponentials, that means only

 * M < A.G(N)

for some absolute constant A. The value of A, the so-called implied constant, may also need to be made explicit, for computational purposes. One reason Landau notation was a popular introduction is that it hides exactly what A is. In some indirect forms of proof it may not be at all obvious that the implied constant can be made explicit.

The 'Siegel period'
Many of the principal results of analytic number theory that were proved in the period 1900–1950 were in fact ineffective. The main examples were:

 * The Thue–Siegel–Roth theorem
 * Siegel's theorem on integral points, from 1929
 * The 1934 theorem of Hans Heilbronn and Edward Linfoot on the class number 1 problem
 * The 1935 result on the Siegel zero
 * The Siegel–Walfisz theorem based on the Siegel zero.

The concrete information that was left theoretically incomplete included lower bounds for class numbers (ideal class groups for some families of number fields grow); and bounds for the best rational approximations to algebraic numbers in terms of denominators. These latter could be read quite directly as results on Diophantine equations, after the work of Axel Thue. The result used for Liouville numbers in the proof is effective in the way it applies the mean value theorem: but improvements (to what is now the Thue–Siegel–Roth theorem) were not.

Later work
Later results, particularly of Alan Baker, changed the position. Qualitatively speaking, Baker's theorems look weaker, but they have explicit constants and can actually be applied, in conjunction with machine computation, to prove that lists of solutions (suspected to be complete) are actually the entire solution set.

Theoretical issues
The difficulties here were met by radically different proof techniques, taking much more care about proofs by contradiction. The logic involved is closer to proof theory than to that of computability theory and computable functions. It is rather loosely conjectured that the difficulties may lie in the realm of computational complexity theory. Ineffective results are still being proved in the shape A or B, where we have no way of telling which.

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

title: Effective results in number theory


text:
Pediculus humanus

Pediculus humanus is a species of louse that infects humans. It comprises two subspecies:
 * Pediculus humanus humanus Linnaeus, 1758 – body louse
 * Pediculus humanus capitis De Geer, 1767 – head louse

title: Pediculus humanus


text:
Ecteninion

Ecteninion is an extinct genus of meat-eating cynodonts that lived during the Late Triassic (Carnian) in South America. The type species Ecteninion lunensis was named by R.N. Martinez, C.L. May, and C.A. Forster in 1996. E. lunensis is known from a nearly complete skull of about 11 cm in length. It was found in the Cancha de Bochas Member of the Ischigualasto Formation in the Ischigualasto-Villa Unión Basin in northwestern Argentina. It has been interpreted as a basal eucynodont. The holotype is in the collection of the Universidad Nacional de San Juan.

Phylogeny
Ecteninion in a cladogram after Hopson & Kitching (2001):

Cladogram after Stefanello et al. (2023):

title: Ecteninion


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"Hahnia" obliqua

"Hahnia" obliqua is a poorly known species of meat-eating stem-mammals (cynodonts) that lived during the Upper Triassic in Europe. It is based on tiny, isolated teeth, and its affinities with other cynodonts are unclear. The generic name is not valid, thus the quotation marks. The name Hahnia has already been used for a spider. The authors are aware of this, and will doubtless come up with a new name sometime in the future.

The genus "Hahnia" ("for Hahn") was named by Pascal Godefroit and Bernard Battail in 1997 based on a single species.

Fossil remains of the species "Hahnia" obliqua have been found in the Norian (late) - Rhaetian (early) (Upper Triassic)-age strata of Saint-Nicolas-de-Port in France. Apart from not yet having a proper name, these teeth look rather boring. The crown slopes backwards and has three cusps, though that's more apparent when seen from above than it is from the side. The largest cusp, the middle one, has a "somewhat blunt" apex. Its two colleagues, which "are not very well separated from the main cusp", are "very blunt", (quotes from Godfroit & Battauk 1997, p. 588). "There is no constriction between the crown and the root."

Boring looking or not, these were nevertheless effective for cutting up small portions of prey. The authors discuss similarities with teeth of galesaurids (something like forerunners of the eucynodonts), Cynognathus, chiniquodontids, tritheledontids, dromatheriids and various other small cynodonts of the European Upper Triassic; They are all carnivores of one size or another. However, as there are also clear differences to the tiny teeth of "Hahnia", the authors plump for Cynodontia incertae sedis (aka of some kind or other).

title: "Hahnia" obliqua


text:
Areola

The human areola (areola mammae, or  ) is the pigmented area on the breast around the nipple. Areola, more generally, is a small circular area on the body with a different histology from the surrounding tissue, or other small circular areas such as an inflamed region of skin.

The mature human female nipple has several small openings arranged radially around the tip of the lactiferous ducts from which milk is released during lactation. Other small openings in the areola are sebaceous glands, also known as areolar glands.

Shade
The areolae can range from pink to red to brown to dark brown or nearly black, but generally tend to be paler among people with lighter skin tones and darker among people with darker skin tones. A reason for the differing color may be to make the nipple area more visible to the infant.

Size and shape
The size and shape of areolae and nipples are also highly variable, with those of women usually being larger than those of men and prepubescent girls. Human areolae are mostly circular in shape, but many women have large areolae that are noticeably elliptical.

The average diameter of male areolae is around 28.0 mm. Sexually mature women have an average of 38.1 mm, but sizes can exceed 100 mm. Lactating women, and women with particularly large breasts, may have even larger areolae. A function of the specialized dermis of the areola is to protect the regular breast skin from wear, cracking, and irritation. Infants sometimes create trauma to the nipple and areolae by latching-on.

Rated according to the Tanner scale of female physical development, the areolae enlarge during stage 3, but they show no separation of contour. During stage 4, the areolae and papillae rise above breast level and form secondary mounds. By stage 5, the breasts have fully developed. As this has resulted in recession of the areolae, the papillae may reach a little above the breasts' contour.

Mechanoreceptors
Breastfeeding by the baby stimulates slowly and rapidly adapting mechanoreceptors that are densely packed around the areolar region.

Diseases
Paget's disease of the breast is a malignant condition that outwardly may have the appearance of eczema, with skin changes involving the areola and nipple.

title: Areola


text:
Christian Bök

Christian Bök, FRSC (born August 10, 1966, in Toronto, Canada) is a Canadian poet known for his experimental works. He is the author of Eunoia, which won the Canadian Griffin Poetry Prize.

Life and work
He was born "Christian Book", but uses "Bök" as a pseudonym.

He began writing seriously in his early twenties, while earning his B.A. and M.A. degrees at Carleton University in Ottawa. He returned to Toronto in the early 1990s to study for a Ph.D. in English literature at York University, where he encountered a burgeoning literary community that included Steve McCaffery, Christopher Dewdney, and Darren Wershler. Since 2004, he taught at the University of Calgary — but as of 2022, he works as an artist in Melbourne, Australia, and he serves as a Professor (Honorary Appointee) at Charles Darwin University in Darwin, Australia.

In 1994, Bök published Crystallography, "a pataphysical encyclopaedia that misreads the language of poetics through the conceits of geology." The Village Voice said of it: "Bök's concise reflections on mirrors, fractals, stones, and ice diabolically change the way you think about language — his, yours — so that what begins as description suddenly seems indistinguishable from the thing itself." Crystallography was reissued in 2003 and was nominated for a Gerald Lampert Award.

Bök is a sound poet, who has performed an extremely condensed version of the "Ursonate" by Kurt Schwitters. He has created conceptual art, making artist's books from Rubik's Cubes and Lego bricks. He has also worked in science-fiction television by constructing artificial languages for Gene Roddenberry's Earth: Final Conflict and Peter Benchley's Amazon.

Eunoia
Bök is most famous for writing Eunoia (2001), a book that took him seven years to finish. Eunoia consists of univocalic lipograms. The book uses only one vowel in each of its five chapters. In the book's main part, each chapter uses just a single vowel, producing sentences such as this: "Enfettered, these sentences repress free speech." Bök believes that "his book proves that each vowel has its own personality."

Edited by Darren Wershler and published by Coach House Books in 2001, Eunoia won the 2002 Griffin Poetry Prize and sold more than 20,000 copies. Canongate published "Eunoia" in Britain in Oct. 2008. The book was also a bestseller there, reaching #8 on the Top 10 bestselling charts for the year.

The Xenotext
The Xenotext is an ongoing work of BioArt which claims to be “the first example of ‘living poetry.’” The Xenotext consists of a single sonnet (called "Orpheus"), which gets translated into a gene and then integrated into a cell, causing the cell to "read" this poem, and in reply, the cell builds a protein — one whose sequence of amino acids encodes yet another sonnet (called "Eurydice"). The cell becomes not only a durable archive for storing a poem, but also an operant machine for writing a poem. The gene has so far worked properly in cultures of E. coli, but the intended symbiote is D. radiodurans ("the dire seed, immune to radiation") — an extremophile, able to thrive in very inhospitable environments, deadly to most life on Earth.

According to Bök from an interview in 2007, the final product will include:

"a poetic manual that showcases the text of the poem, followed by an artfully designed monograph about the experiment, including, for example, the chemical alphabet for the cipher, the genetic sequence for the poetry, the schematics for the protein, and even a photograph of the microbe, complete with other apparati, such as charts, graphs, images, and essays, all outlining our results."

Bök has collaborated with laboratories at the University of Calgary, the University of Wyoming, and the University of Texas (Austin), to realize his design. In 2011, nine years after conceiving The Xenotext, Bök announced that labs had performed a successful test run of his “poetic cipher,” meaning that:

"when implanted in the genome of [the] bacterium, [the] poem (which begins ‘any style of life/ is prim…’) does in fact cause the bacterium to write, in response, its own poem (which begins ‘the faery is rosy/ of glow…’)."

In 2015, The Xenotext: Book I was published — a work consisting of meditations on both science and poetics (addressing their mythic drives for immortality). This first, "Orphic" volume sets the conceptual groundwork for the second, "Eurydicean" volume, which will document the experiment itself. The Xenotext: Book II remains forthcoming.

The Kazimir Effect
Since 2017, Bök has been working on a lengthy project of visual poetry, inspired by Suprematist Composition: White on White by Kazimir Malevich. This project has culminated in the publication of a book entitled The Kazimir Effect (Penteract Press, 2021), listed as one of the Times Literary Supplement’s Books of the Year 2021.

Recognition
Eunoia won the Griffin Poetry Prize in 2002.

Bök's poem "Vowels" was used in the lyrics of a song on the EP A Quick Fix of Melancholy (2003) by the Norwegian band Ulver.

In 2006, Christian Bök and his work were the subject of an episode of the television series Heart of a Poet, produced by Canadian filmmaker Maureen Judge.

On May 31, 2011, The BBC World Service broadcast Bök reading "The Xenotext."

title: Christian Bök


text:
Greeks (finance)

In mathematical finance, the Greeks are the quantities (known in calculus as partial derivatives; first-order or higher) representing the sensitivity of the price of a derivative instrument such as an option to changes in one or more underlying parameters on which the value of an instrument or portfolio of financial instruments is dependent. The name is used because the most common of these sensitivities are denoted by Greek letters (as are some other finance measures). Collectively these have also been called the risk sensitivities, risk measures or hedge parameters.

Use of the Greeks
The Greeks are vital tools in risk management. Each Greek measures the sensitivity of the value of a portfolio to a small change in a given underlying parameter, so that component risks may be treated in isolation, and the portfolio rebalanced accordingly to achieve a desired exposure; see for example delta hedging.

The Greeks in the Black–Scholes model (a relatively simple idealised model of certain financial markets) are relatively easy to calculate — a desirable property of financial models — and are very useful for derivatives traders, especially those who seek to hedge their portfolios from adverse changes in market conditions. For this reason, those Greeks which are particularly useful for hedging—such as delta, theta, and vega—are well-defined for measuring changes in the parameters spot price, time and volatility. Although rho (the partial derivative with respect to the risk-free interest rate) is a primary input into the Black–Scholes model, the overall impact on the value of a short-term option corresponding to changes in the risk-free interest rate is generally insignificant and therefore higher-order derivatives involving the risk-free interest rate are not common.

The most common of the Greeks are the first order derivatives: delta, vega, theta and rho; as well as gamma, a second-order derivative of the value function. The remaining sensitivities in this list are common enough that they have common names, but this list is by no means exhaustive.

The players in the market make competitive trades involving many billions (of $, £ or €) of underlying every day, so it is important to get the sums right. In practice they will use more sophisticated models which go beyond the simplifying assumptions used in the Black-Scholes model and hence in the Greeks.

Names
The use of Greek letter names is presumably by extension from the common finance terms alpha and beta, and the use of sigma (the standard deviation of logarithmic returns) and tau (time to expiry) in the Black–Scholes option pricing model. Several names such as "vega" (whose symbol is similar to the lower-case Greek letter nu; the use of that name might have led to confusion) and "zomma" are invented, but sound similar to Greek letters. The names "color" and "charm" presumably derive from the use of these terms for exotic properties of quarks in particle physics.

Delta
Delta, $$\Delta$$, measures the rate of change of the theoretical option value with respect to changes in the underlying asset's price. Delta is the first derivative of the value $$V$$ of the option with respect to the underlying instrument's price $$S$$.

 * $$\Delta = \frac{\partial V}{\partial S}$$

Practical use
For a vanilla option, delta will be a number between 0.0 and 1.0 for a long call (or a short put) and 0.0 and −1.0 for a long put (or a short call); depending on price, a call option behaves as if one owns 1 share of the underlying stock (if deep in the money), or owns nothing (if far out of the money), or something in between, and conversely for a put option. The difference between the delta of a call and the delta of a put at the same strike is equal to one. By put–call parity, long a call and short a put is equivalent to a forward F, which is linear in the spot S, with unit factor, so the derivative dF/dS is 1. See the formulas below.

These numbers are commonly presented as a percentage of the total number of shares represented by the option contract(s). This is convenient because the option will (instantaneously) behave like the number of shares indicated by the delta. For example, if a portfolio of 100 American call options on XYZ each have a delta of 0.25 (= 25%), it will gain or lose value just like 2,500 shares of XYZ as the price changes for small price movements (100 option contracts covers 10,000 shares). The sign and percentage are often dropped – the sign is implicit in the option type (negative for put, positive for call) and the percentage is understood. The most commonly quoted are 25 delta put, 50 delta put/50 delta call, and 25 delta call. 50 Delta put and 50 Delta call are not quite identical, due to spot and forward differing by the discount factor, but they are often conflated.

Delta is always positive for long calls and negative for long puts (unless they are zero). The total delta of a complex portfolio of positions on the same underlying asset can be calculated by simply taking the sum of the deltas for each individual position – delta of a portfolio is linear in the constituents. Since the delta of underlying asset is always 1.0, the trader could delta-hedge his entire position in the underlying by buying or shorting the number of shares indicated by the total delta. For example, if the delta of a portfolio of options in XYZ (expressed as shares of the underlying) is +2.75, the trader would be able to delta-hedge the portfolio by selling short 2.75 shares of the underlying. This portfolio will then retain its total value regardless of which direction the price of XYZ moves. (Albeit for only small movements of the underlying, a short amount of time and not-withstanding changes in other market conditions such as volatility and the rate of return for a risk-free investment).

As a proxy for probability
The (absolute value of) Delta is close to, but not identical with, the percent moneyness of an option, i.e., the implied probability that the option will expire in-the-money (if the market moves under Brownian motion in the risk-neutral measure). For this reason some option traders use the absolute value of delta as an approximation for percent moneyness. For example, if an out-of-the-money call option has a delta of 0.15, the trader might estimate that the option has approximately a 15% chance of expiring in-the-money. Similarly, if a put contract has a delta of −0.25, the trader might expect the option to have a 25% probability of expiring in-the-money. At-the-money calls and puts have a delta of approximately 0.5 and −0.5 respectively with a slight bias towards higher deltas for ATM calls since the risk-free rate introduces some offset to the delta. The actual probability of an option finishing in the money is its dual delta, which is the first derivative of option price with respect to strike.

Relationship between call and put delta
Given a European call and put option for the same underlying, strike price and time to maturity, and with no dividend yield, the sum of the absolute values of the delta of each option will be 1 – more precisely, the delta of the call (positive) minus the delta of the put (negative) equals 1. This is due to put–call parity: a long call plus a short put (a call minus a put) replicates a forward, which has delta equal to 1.

If the value of delta for an option is known, one can calculate the value of the delta of the option of the same strike price, underlying and maturity but opposite right by subtracting 1 from a known call delta or adding 1 to a known put delta.

 * $$\Delta(\text{call}) - \Delta(\text{put}) = 1, \text{ therefore: } \Delta(\text{call}) = \Delta(\text{put}) + 1 \text{ and } \Delta(\text{put}) = \Delta(\text{call}) - 1. $$

For example, if the delta of a call is 0.42 then one can compute the delta of the corresponding put at the same strike price by 0.42 − 1 = −0.58. To derive the delta of a call from a put, one can similarly take −0.58 and add 1 to get 0.42.

Vega
Vega measures sensitivity to volatility. Vega is the derivative of the option value with respect to the volatility of the underlying asset.

 * $$\mathcal{V}=\frac{\partial V}{\partial \sigma}$$

Vega is not the name of any Greek letter. The glyph used is a non-standard majuscule version of the Greek letter nu ($\nu$ ), written as $$\mathcal{V}$$. Presumably the name vega was adopted because the Greek letter nu looked like a Latin v|vee, and vega was derived from vee by analogy with how beta, eta, and theta are pronounced in American English.

The symbol kappa, $$\kappa$$, is sometimes used (by academics) instead of vega (as is tau ($$\tau$$) or capital lambda ($$\Lambda$$),

though these are rare).

Vega is typically expressed as the amount of money per underlying share that the option's value will gain or lose as volatility rises or falls by 1 percentage point. All options (both calls and puts) will gain value with rising volatility.

Vega can be an important Greek to monitor for an option trader, especially in volatile markets, since the value of some option strategies can be particularly sensitive to changes in volatility. The value of an at-the-money option straddle, for example, is extremely dependent on changes to volatility. See Volatility risk.

Theta
Theta, $$\Theta$$, measures the sensitivity of the value of the derivative to the passage of time (see Option time value): the "time decay."

 * $$\Theta = -\frac{\partial V}{\partial \tau}$$

As time passes, with decreasing time to expiry and all else being equal, an option's extrinsic value decreases. Typically (but see below), this means an option loses value with time, which is conventionally referred to as long options typically having short (negative) theta. In fact, typically, the literal first derivative w.r.t. time of an option's value is a positive number. The change in option value is typically negative because the passage of time is a negative number (a decrease to $$ \tau \, $$, time to expiry). However, by convention, practitioners usually prefer to refer to theta exposure ("decay") of a long option as negative (instead of the passage of time as negative), and so theta is usually reported as -1 times the first derivative, as above.

While extrinsic value is decreasing with time passing, sometimes a countervailing factor is discounting. For deep-in-the-money options of some types (for puts in Black-Scholes, puts and calls in Black's), as discount factors increase towards 1 with the passage of time, that is an element of increasing value in a long option. Sometimes deep-in-the-money options will gain more from increasing discount factors than they lose from decreasing extrinsic value, and reported theta will be a positive value for a long option instead of a more typical negative value (and the option will be an early exercise candidate, if exercisable, and a European option may become worth less than parity).

By convention in options valuation formulas, $$ \tau \, $$, time to expiry, is defined in years. Practitioners commonly prefer to view theta in terms of change in number of days to expiry rather than number of years to expiry. Therefore, reported theta is usually divided by number of days in a year. (Whether to count calendar days or business days varies by personal choice, with arguments for both.)

Rho
Rho, $$\rho$$, measures sensitivity to the interest rate: it is the derivative of the option value with respect to the risk-free interest rate (for the relevant outstanding term).

 * $$\rho = \frac{\partial V}{\partial r}$$

Except under extreme circumstances, the value of an option is less sensitive to changes in the risk-free interest rate than to changes in other parameters. For this reason, rho is the least used of the first-order Greeks.

Rho is typically expressed as the amount of money, per share of the underlying, that the value of the option will gain or lose as the risk-free interest rate rises or falls by 1.0% per annum (100 basis points).

Lambda
Lambda, $$\lambda$$, omega, $$\Omega$$, or elasticity is the percentage change in option value per percentage change in the underlying price, a measure of leverage, sometimes called gearing.

 * $$\lambda = \Omega = \frac{\partial V}{\partial S}\times\frac{S}{V}$$

It holds that $$\lambda = \Omega = \Delta\times\frac{S}{V}$$.

It is similar to the concept of delta but expressed in percentage terms rather than absolute terms.

Epsilon
Epsilon, $$\varepsilon$$ (also known as psi, $$\psi$$), is the percentage change in option value per percentage change in the underlying dividend yield, a measure of the dividend risk. The dividend yield impact is in practice determined using a 10% increase in those yields. Obviously, this sensitivity can only be applied to derivative instruments of equity products.

 * $$\varepsilon = \psi = \frac{\partial V}{\partial q}$$

Numerically, all first-order sensitivities can be interpreted as spreads in expected returns. Information geometry offers another (trigonometric) interpretation.

Gamma
Gamma, $$\Gamma$$, measures the rate of change in the delta with respect to changes in the underlying price. Gamma is the second derivative of the value function with respect to the underlying price.

 * $$\Gamma = \frac{\partial \Delta}{\partial S} = \frac{\partial^2 V}{\partial S^2}$$

Most long options have positive gamma and most short options have negative gamma. Long options have a positive relationship with gamma because as price increases, Gamma increases as well, causing Delta to approach 1 from 0 (long call option) and 0 from −1 (long put option). The inverse is true for short options. Gamma is greatest approximately at-the-money (ATM) and diminishes the further out you go either in-the-money (ITM) or out-of-the-money (OTM). Gamma is important because it corrects for the convexity of value.

When a trader seeks to establish an effective delta-hedge for a portfolio, the trader may also seek to neutralize the portfolio's gamma, as this will ensure that the hedge will be effective over a wider range of underlying price movements.

Vanna
Vanna, also referred to as DvegaDspot and DdeltaDvol, is a second-order derivative of the option value, once to the underlying spot price and once to volatility. It is mathematically equivalent to DdeltaDvol, the sensitivity of the option delta with respect to change in volatility; or alternatively, the partial of vega with respect to the underlying instrument's price. Vanna can be a useful sensitivity to monitor when maintaining a delta- or vega-hedged portfolio as vanna will help the trader to anticipate changes to the effectiveness of a delta-hedge as volatility changes or the effectiveness of a vega-hedge against change in the underlying spot price.

If the underlying value has continuous second partial derivatives, then $$\text{Vanna} = \frac{\partial \Delta}{\partial \sigma} = \frac{\partial \mathcal{V}}{\partial S} = \frac{\partial^2 V}{\partial S \, \partial \sigma}. $$

Charm
Charm or delta decay measures the instantaneous rate of change of delta over the passage of time.

 * $$\text{Charm} = -\frac{\partial \Delta}{\partial \tau} = \frac{\partial \Theta}{\partial S} = -\frac{\partial^2 V}{\partial \tau \, \partial S}$$

Charm has also been called DdeltaDtime. Charm can be an important Greek to measure/monitor when delta-hedging a position over a weekend. Charm is a second-order derivative of the option value, once to price and once to the passage of time. It is also then the derivative of theta with respect to the underlying's price.

The mathematical result of the formula for charm (see below) is expressed in delta/year. It is often useful to divide this by the number of days per year to arrive at the delta decay per day. This use is fairly accurate when the number of days remaining until option expiration is large. When an option nears expiration, charm itself may change quickly, rendering full day estimates of delta decay inaccurate.

Vomma
Vomma, volga, vega convexity, or DvegaDvol measures second-order sensitivity to volatility. Vomma is the second derivative of the option value with respect to the volatility, or, stated another way, vomma measures the rate of change to vega as volatility changes.

 * $$\text{Vomma} = \frac{\partial \mathcal{V}}{\partial \sigma} = \frac{\partial^2 V}{\partial \sigma^2}$$

With positive vomma, a position will become long vega as implied volatility increases and short vega as it decreases, which can be scalped in a way analogous to long gamma. And an initially vega-neutral, long-vomma position can be constructed from ratios of options at different strikes. Vomma is positive for long options away from the money, and initially increases with distance from the money (but drops off as vega drops off). (Specifically, vomma is positive where the usual d1 and d2 terms are of the same sign, which is true when d1 < 0 or d2 > 0.)

Veta
Veta, vega decay or DvegaDtime measures the rate of change in the vega with respect to the passage of time. Veta is the second derivative of the value function; once to volatility and once to time.

 * $$\text{Veta} = \frac{\partial \mathcal{V}}{\partial \tau} = \frac{\partial^2 V}{\partial \sigma \, \partial \tau}$$

It is common practice to divide the mathematical result of veta by 100 times the number of days per year to reduce the value to the percentage change in vega per one day.

Vera
Vera (sometimes rhova) measures the rate of change in rho with respect to volatility. Vera is the second derivative of the value function; once to volatility and once to interest rate.

 * $$\text{Vera} = \frac{\partial \rho}{\partial \sigma} = \frac{\partial^2 V}{\partial \sigma \, \partial r}$$

The word 'Vera' was coined by R. Naryshkin in early 2012 when this sensitivity needed to be used in practice to assess the impact of volatility changes on rho-hedging, but no name yet existed in the available literature. 'Vera' was picked to sound similar to a combination of Vega and Rho, its respective first-order Greeks. This name is now in a wider use, including, for example, the Maple computer algebra software (which has 'BlackScholesVera' function in its Finance package).

Second-order partial derivative with respect to strike K
This partial derivative has a fundamental role in the Breeden–Litzenberger formula, which uses quoted call option prices to estimate the risk-neutral probabilities implied by such prices.

 * $$\varpi =\frac{\partial^2 V}{\partial K^2}$$

For call options, it can be approximated using infinitesimal portfolios of butterfly strategies.

Speed
Speed measures the rate of change in Gamma with respect to changes in the underlying price.

 * $$\text{Speed} = \frac{\partial\Gamma}{\partial S} = \frac{\partial^3 V}{\partial S^3}$$

This is also sometimes referred to as the gamma of the gamma or DgammaDspot. Speed is the third derivative of the value function with respect to the underlying spot price. Speed can be important to monitor when delta-hedging or gamma-hedging a portfolio.

Zomma
Zomma measures the rate of change of gamma with respect to changes in volatility.

 * $$\text{Zomma} = \frac{\partial \Gamma}{\partial \sigma} = \frac{\partial \text{vanna}}{\partial S} = \frac{\partial^3 V}{\partial S^2 \, \partial \sigma}$$

Zomma has also been referred to as DgammaDvol. Zomma is the third derivative of the option value, twice to underlying asset price and once to volatility. Zomma can be a useful sensitivity to monitor when maintaining a gamma-hedged portfolio as zomma will help the trader to anticipate changes to the effectiveness of the hedge as volatility changes.

Color
Color, gamma decay or DgammaDtime measures the rate of change of gamma over the passage of time.

 * $$\text{Color} = \frac{\partial \Gamma}{\partial \tau} = \frac{\partial^3 V}{\partial S^2 \, \partial \tau}$$

Color is a third-order derivative of the option value, twice to underlying asset price and once to time. Color can be an important sensitivity to monitor when maintaining a gamma-hedged portfolio as it can help the trader to anticipate the effectiveness of the hedge as time passes.

The mathematical result of the formula for color (see below) is expressed in gamma per year. It is often useful to divide this by the number of days per year to arrive at the change in gamma per day. This use is fairly accurate when the number of days remaining until option expiration is large. When an option nears expiration, color itself may change quickly, rendering full day estimates of gamma change inaccurate.

Ultima
Ultima measures the sensitivity of the option vomma with respect to change in volatility.
 * $$\text{Ultima}= \frac{\partial \text{vomma}}{\partial \sigma} = \frac{\partial^3 V}{\partial \sigma^3}$$

Ultima has also been referred to as DvommaDvol. Ultima is a third-order derivative of the option value to volatility.

Greeks for multi-asset options
If the value of a derivative is dependent on two or more underlyings, its Greeks are extended to include the cross-effects between the underlyings.

Correlation delta measures the sensitivity of the derivative's value to a change in the correlation between the underlyings. It is also commonly known as cega.

Cross gamma measures the rate of change of delta in one underlying to a change in the level of another underlying.

Cross vanna measures the rate of change of vega in one underlying due to a change in the level of another underlying. Equivalently, it measures the rate of change of delta in the second underlying due to a change in the volatility of the first underlying.

Cross volga measures the rate of change of vega in one underlying to a change in the volatility of another underlying.

Formulae for European option Greeks
The Greeks of European options (calls and puts) under the Black–Scholes model are calculated as follows, where $$\varphi$$ (phi) is the standard normal probability density function and $$\Phi$$ is the standard normal cumulative distribution function. Note that the gamma and vega formulas are the same for calls and puts.

For a given:
 * Stock price $$ S \, $$,
 * Strike price $$ K \, $$,
 * Risk-free rate $$ r \, $$,
 * Annual dividend yield $$ q \, $$,
 * Time to maturity $$ \tau = T - t \, $$ (represented as a unit-less fraction of one year), and
 * Volatility $$ \sigma \, $$.

where

 * $$\begin{align}

d_1 &= \frac{\ln(S/K) + \left(r - q + \frac{1}{2}\sigma^2\right)\tau}{\sigma\sqrt{\tau}} \\ d_2 &= \frac{\ln(S/K) + \left(r - q - \frac{1}{2}\sigma^2\right)\tau}{\sigma\sqrt{\tau}} = d_1 - \sigma\sqrt{\tau} \\ \varphi(x) &= \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2} x^2} \\ \Phi(x) &= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-\frac{1}{2} y^2} \,dy = 1 - \frac{1}{\sqrt{2\pi}} \int_x^\infty e^{-\frac{1}{2} y^2} \,dy \end{align}$$

Under the Black model (commonly used for commodities and options on futures) the Greeks can be calculated as follows:

where

 * $$\begin{align}

d_1 &= \frac{\ln(F/K) + \frac{1}{2}\sigma^2\tau}{\sigma\sqrt\tau} \\ d_2 &= \frac{\ln(F/K) - \frac{1}{2}\sigma^2\tau}{\sigma\sqrt\tau} = d_1 - \sigma\sqrt\tau \end{align}$$

(*) It can be shown that $$ F\varphi(d_1) = K\varphi(d_2) $$

Micro proof: let $$ x= \sigma \sqrt \tau $$

$$ d_1 = \frac{\ln \frac F K + \frac 1 2 x^2}{x}$$

$$ d_1\cdot x = \ln \frac F K + \frac 1 2 x^2$$

$$ \ln(F/K) = d_1\cdot x - \frac{1}{2}x^2$$

$$ \frac{F}{K} = e^{d_1 \cdot x - \frac{1}{2}x^2}$$

Then we have: $$ \frac{F}{K} \cdot \frac{\varphi(d_1)}{\varphi(d_2)} = \frac{F}{K} \cdot e^{\frac{1}{2}\cdot {d_2}^2 - \frac{1}{2}\cdot{d_1}^2}$$

 * $$ =e^{d_1 x - \frac{1}{2}x^2} \cdot e^{\frac{1}{2}\cdot{(d_1-x)}^2 - \frac{1}{2}\cdot{d_1}^2} = e^{d_1 x - \frac{1}{2}x^2 + \frac{1}{2} \cdot (2d_1 - x)(-x)} = e^0 = 1.$$

So $$ F\varphi(d_1) = K\varphi(d_2) $$

Related measures
Some related risk measures of financial instruments are listed below.

Bond duration and convexity
In trading bonds and other fixed income securities, various measures of bond duration are used analogously to the delta of an option. The closest analogue to the delta is DV01, which is the reduction in price (in currency units) for an increase of one basis point (i.e. 0.01% per annum) in the yield, where yield is the underlying variable; see. (Related is CS01, measuring sensitivity to credit spread.)

Analogous to the lambda is the modified duration, which is the percentage change in the market price of the bond(s) for a unit change in the yield (i.e. it is equivalent to DV01 divided by the market price). Unlike the lambda, which is an elasticity (a percentage change in output for a percentage change in input), the modified duration is instead a semi-elasticity—a percentage change in output for a unit change in input. See also Key rate duration.

Bond convexity is a measure of the sensitivity of the duration to changes in interest rates, the second derivative of the price of the bond with respect to interest rates (duration is the first derivative); it is then analogous to gamma. In general, the higher the convexity, the more sensitive the bond price is to the change in interest rates. Bond convexity is one of the most basic and widely used forms of convexity in finance.

For a bond with an embedded option, the standard yield to maturity based calculations here do not consider how changes in interest rates will alter the cash flows due to option exercise. To address this, effective duration and effective convexity are introduced. These values are typically calculated using a tree-based model, built for the entire yield curve (as opposed to a single yield to maturity), and therefore capturing exercise behavior at each point in the option's life as a function of both time and interest rates; see.

Beta
The beta (β) of a stock or portfolio is a number describing the volatility of an asset in relation to the volatility of the benchmark that said asset is being compared to. This benchmark is generally the overall financial market and is often estimated via the use of representative indices, such as the S&P 500.

An asset has a Beta of zero if its returns change independently of changes in the market's returns. A positive beta means that the asset's returns generally follow the market's returns, in the sense that they both tend to be above their respective averages together, or both tend to be below their respective averages together. A negative beta means that the asset's returns generally move opposite the market's returns: one will tend to be above its average when the other is below its average.

Fugit
The fugit is the expected time to exercise an American or Bermudan option. Fugit is usefully computed for hedging purposes — for example, one can represent flows of an American swaption like the flows of a swap starting at the fugit multiplied by delta, and then use these to compute other sensitivities.

title: Greeks (finance)


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Runabout (boat)

A runabout is any small motorboat holding between four and eight people, well suited to moving about on the water. Characteristically between 20' and 35' in length, runabouts are used for pleasure activities like boating, fishing, and water skiing, as a ship's tender for larger vessels, or in racing. Some common runabout types are bow rider, center console, cuddy boat and walkaround. The world's largest runabout, Pardon Me, is 48 feet long and owned by the Antique Boat Museum in Clayton, New York.

History
The first runabouts date back to the 1920s and were originally small, fast, powerful, varnished, wooden boats created to take advantage of the power of outboard motors such as the first Evinrude, introduced in 1909.

In order to gain speed, the hull shape had to be designed to take advantage of hydroplaning; a hydrofoil-like design would allow the boat to skim atop the water's surface at high speed instead of needing to push aside large quantities of water to move forward. Another design change which followed soon after was the replacement of the tiller and rudder control with a rudder controlled by a steering wheel, allowing the operator a comfortable forward-facing position. A remote lever to allow the engines to be placed into a reverse gear was another early innovation.

Among the leading builders of 1920s runabouts was John L. Hacker, who founded the Hacker Boat Company in 1908. Hacker was a pioneering naval architect who developed many design innovations, like the 'V-bottom'. His designs became the model upon which virtually all subsequent runabouts were based.

Shortly, similar upscale varnished-wood runabouts by Gar Wood and Chris-Craft and were also available, fitted with windshields to protect the cockpits and up to 400 hp Liberty V-12 marinized surplus World War I aero engines built for speed.

But by the late 1940s, Gar Wood had stopped producing boats, and by the 1960s Chris-Craft was moving to the more modern materials of plastic and fiberglass. Hackercraft, with multiple changes in ownership, continued on.

The mahogany runabouts built by Italian builder Carlo Riva in the late 1950s and the 1960s are considered by many to be premier European examples of the type. The most famous Riva of all time was the Carlo Riva design called the Aquarama Special.

Construction and materials
Originally, runabouts were made entirely of wood, with mahogany used for hulls and planking and oak for framing. The use of aluminium in small boat construction came soon after World War II because of availability of aircraft materials as war surplus. Fiberglass was then introduced as another way to reduce the maintenance, cost and weight of watercraft. Given the cost benefits and personal enjoyment of boat building, do-it-yourself ′Kit Boats′ were also introduced using plywood material. In 1955, Chris-Craft created The Plywood Boat Division which marketed both Kit and pre-built plywood craft.

By 1960, wooden powerboats had become rare since most new vessels used fiberglass or other lightweight materials, including fiber reinforced plastic materials to reduce weight and maximize speed, particularly in racing craft. The art of boatbuilding in wood has been largely lost since it requires a level of craftsmanship impossible in large scale production boat building. One exception is the Hacker Boat Company, which continues to produce mahogany boats on the shores of Lake George, New York. Other wooden boatbuilders include Graf, J-Craft, and Boesch.

Propulsion
Runabouts can be powered by inboard engines, outboards, jet drives, or inboard-outboard (I/O) drives. Engines can be gasoline or diesel systems.

Inboards have the engine block permanently mounted within the hull of the boat, with a drive shaft and a propeller to drive the craft underneath the hull, and a separate rudder to steer the craft.

Outboards are steerable external drive motors containing the engine block, linkage gears, and propeller within a single unit, taking the place of a rudder. Outboard drives are mounted to the transom and steered by a remote system leading to a wheel mounted on the boat's console.

Inboard-Outboard (or stern drives) are a hybrid, with an engine block mounted within the hull linked to a pivotable lower drive unit which steers the craft, similar to an outboard motor.

Jet Drives have a propeller enclosed in a pump-jet that draws water from underneath the hull and expels it through a swiveling nozzle in the stern. They are highly maneuverable and tolerant of shallow water, but need larger engines and use more fuel than the other alternatives.

title: Runabout (boat)