Raphael Haftka

Raphael T. Haftka (February 22, 1944 – August 16, 2020) was an American engineer, a Distinguished Professor at University of Florida and an Elected Fellow of the American Institute of Aeronautics and Astronautics.
Born February 22, 1944, in Tel-Aviv, Israel. Education: Ph.D. 1971, Department of Aerospace and Mechanical Engineering Sciences, the University of California at San
Diego, B.Sc. 1965, M.Sc. 1968, Aeronautical Engineering, Technion-Israel Institute of Technology. His professional journey started as
an aerodynamicist at the Israeli Aircraft Industries from 1965 to 1968. From 1973 to 1975, he was a senior lecturer at Technion-Israel Institute of Technology, after postdoctoral research fellowships at NASA and other US organizations. His career moved to the USA when he became an Assistant Professor at the Illinois Institute of Technology in 1975. Subsequently, he held the positions of Professor/ Christopher Kraft Professor at Virginia Tech from 1981 to 1994 and Professor/Distinguished Professor
at the University of Florida from 1995 until retirement in 2019.
Professor Haftka's contributions to our research field are immeasurable—some important statistics: 330 journal papers, 530 conference proceedings papers, 12 book chapters, two textbooks, and over 34,000 Google citations. His major research impacts include—early pioneering contributions to the approximation concept, including the far-reaching convex/conservative approximation, landmark papers on sensitivity analysis, leadership in creating/promoting MDO, surrogate modeling, and nondeterministic methods. Beyond his remarkable contributions as a researcher, Professor Haftka was also a key influencer within his research community. He was among the founders of ISSMO (International Society of Structural and Multidisciplinary Optimization) in 1991 and served as ISSMO’s second president between 1995 and 1999. He has received many distinctions throughout his
including Virginia Tech Alumni Award for Excellence in Research, AIAA Fellow, AIAA Multidisciplinary Design Optimization Award.


Azimuth recording

Azimuth recording is the use of a variation in angle between two recording heads that are recording data so close together on magnetic tape that crosstalk would otherwise likely occur. Normally, the head is perpendicular to the movement of the tape, and this is considered zero degrees. However, if the heads are mounted at slightly different angles (such as ±7 degrees in VHS), destructive interference will occur at high frequencies when reading data recorded in the cross-talking channel but not in the channel that is intended to be read. At low frequencies relative to the maximum allowed by the head gap, however, this technique is ineffective. Thus one head is slanted slightly leftwards and the magnetic gap of the other head slanted slightly rightwards.
To look at it another way, channel A sees the channel B data stretched out in time, hence the technique has a low-pass effect on noise intruding from another channel.
Every videotape system was designed to put as much video as possible onto a given-sized tape, but information from one recording track (pass of the video head) must not interfere with information on adjacent stripes. Using slant azimuth recording, the need for guard bands, that is the blank space between tracks, is eliminated, allowing more recording to be placed on a given length of tape.
All the early low-end reel-to-reel VTR machines and the first VCR cassette formats, the Philips and the Sony U-matic, used this system. Later, the JVC VHS and the Sony Betamax used slant azimuth recording also. These digital VTR versions used azimuth recording as well.


Semmerling LM4

The Semmerling LM4 is a five-shot, manually repeating double-action pocket pistol.
Concept and History.
The Semmerling series of pistols included the LM1, LM2, LM3, LM4 and XLM. The only version available to the public was the LM4, which was first designed and manufactured in the US in the early 1980s and marketed at a price of US$645. The "LM" in the model numbers stands for "Lichtman Model", after the inventor of the design, Philip R. Lichtman. Lichtman applied for and received at least four United States Patents related to the design.
The goal of the LM4 was to achieve a combination of the highest practical firepower in as small and light a gun as possible, so as to be easily concealable. This was approached by selecting the service caliber of .45 ACP for its proven stopping power and opting for a manual repeating mechanism to minimize bulk and weight and ensure flawless operation in the most difficult situations. Because of this the LM4 was first offered to the U.S. Army and government agencies.
The Semmerling LM4 is simple, having only 33 parts (including screws). The original design called for every part except for the springs to be made of S-7 tool steel. Every pistol was Magnafluxed (a method of testing ferrous metals for surface and subsurface flaws) twice in their assembly.
Each pistol was essentially handmade to order with production averaging a little over ten pieces a month. Of the LM4 model, only about 600 were ever produced.
Although the Semmerling was intended as a back-up pistol for those most at risk of having their primary weapon taken from them, some adopted the pistol as a "hide-out" weapon. It is capable of a "shot-a-second" firing rate. With the magazine loaded and a round in the chamber, the Semmerling LM4 is capable of carrying five .45 ACP rounds.
During 2005, the largest collections of Semmerlings comprising examples of the LM1, LM2, LM3, LM4 and the XLM along with all prototypes, tooling, drawings, production and testing equipment, advertising materials and leather accessories were provided to the National Firearms Museum, the Cody Firearms Museum and the Smithsonian. The XLM was a military contract for an automatic version (not manually operated), making the Semmerling XLM the smallest .45 ACP semi-automatic pistol ever produced.
A stainless steel version, not part of the Semmerling Corporation's efforts, has been available at request from the American Derringer corporation since about 1995.
Phillip Lichtman died on June 28, 2017.
Finish and accessories.
The original standard color was matte black with highly polished slide frames. A complete high polish blued finish or a hard chrome plating were optional. A factory presentation model was produced with a high polished gold plated slide.
Operation.
The Semmerling fires from a static, fully locked breech, producing considerable recoil. The barrel mechanism is manually pulled forward to eject the fired round, then pushed back to chamber the next round. This can be accomplished by thumb. For carry, the slide is locked in the closed position by means of a tiny lever located in the middle of the weapon. By carefully pulling the trigger back approx. 1/4 inch and pressing on the lever, the lever clicks into "lock" mode. Hold the lever down and release the trigger. The Semmerling's action is now locked closed with a round in the chamber. To fire, the trigger is pulled back fully. When there is a cartridge in the chamber, the extractor protrudes 1/8 of an inch, so as to act as loaded chamber indicator. It has no safety, decocker, or slide release.
The number of remaining rounds could be seen by using the "thin grip" kit or by removing the grips when carrying the LM4.


Hasselblad E-mount


Saddle-node bifurcation

In the mathematical area of bifurcation theory a saddle-node bifurcation, tangential bifurcation or fold bifurcation is a local bifurcation in which two fixed points (or equilibria) of a dynamical system collide and annihilate each other. The term 'saddle-node bifurcation' is most often used in reference to continuous dynamical systems. In discrete dynamical systems, the same bifurcation is often instead called a fold bifurcation. Another name is blue sky bifurcation in reference to the sudden creation of two fixed points.
If the phase space is one-dimensional, one of the equilibrium points is unstable (the saddle), while the other is stable (the node).
Saddle-node bifurcations may be associated with hysteresis loops and catastrophes.
Normal form.
A typical example of a differential equation with a saddle-node bifurcation is:
Here formula_2 is the state variable and formula_3 is the bifurcation parameter.
In fact, this is a normal form of a saddle-node bifurcation. A scalar differential equation formula_9 which has a fixed point at formula_10 for formula_11 with formula_12 is locally topologically equivalent to formula_13, provided it satisfies formula_14 and formula_15. The first condition is the nondegeneracy condition and the second condition is the transversality condition.
Example in two dimensions.
An example of a saddle-node bifurcation in two dimensions occurs in the two-dimensional dynamical system:
As can be seen by the animation obtained by plotting phase portraits by varying the parameter formula_18,
A saddle-node bifurcation also occurs in the consumer equation (see transcritical bifurcation) if the consumption term is changed from formula_22 to formula_23, that is, the consumption rate is constant and not in proportion to resource formula_2.
Other examples are in modelling biological switches. Recently, it was shown that under certain conditions, the Einstein field equations of General Relativity have the same form as a fold bifurcation. A non-autonomous version of the saddle-node bifurcation (i.e. the parameter is time-dependent) has also been studied.


Minolta V-mount


Avalanche breakdown

Avalanche breakdown (or “the avalanche effect”) is a phenomenon that can occur in both insulating and semiconducting materials. It is a form of electric current multiplication that can allow very large currents within materials which are otherwise good insulators. It is a type of electron avalanche. The avalanche process occurs when carriers in the transition region are accelerated by the electric field to energies sufficient to create mobile or free electron-hole pairs via collisions with bound electrons.
Explanation.
Materials conduct electricity if they contain mobile charge carriers. There are two types of charge carriers in a semiconductor: free electrons (mobile electrons) and electron holes (mobile holes which are missing electrons from the normally occupied electron states). A normally bound electron (e.g., in a bond) in a reverse-biased diode may break loose due to a thermal fluctuation or excitation, creating a mobile electron-hole pair. If there is a voltage gradient (electric field) in the semiconductor, the electron will move towards the positive voltage while the hole will move towards the negative voltage. Usually, the electron and hole will simply move to opposite ends of the crystal and enter the appropriate electrodes. When the electric field is strong enough, the mobile electron or hole may be accelerated to high enough speeds to knock other bound electrons free, creating more free charge carriers, increasing the current and leading to further "knocking out" processes and creating an avalanche. In this way, large portions of a normally insulating crystal can begin to conduct.
The large voltage drop and possibly large current during breakdown necessarily leads to the generation of heat. Therefore, a diode placed into a reverse blocking power application will usually be destroyed by breakdown if the external circuit allows a large current.
In principle, avalanche breakdown only involves the passage of electrons and need not cause damage to the crystal. Avalanche diodes (commonly encountered as high voltage Zener diodes) are constructed to break down at a uniform voltage and to avoid current crowding during breakdown. These diodes can indefinitely sustain a moderate level of current during breakdown.
The voltage at which the breakdown occurs is called the breakdown voltage. There is a hysteresis effect; once avalanche breakdown has occurred, the material will continue to conduct even if the voltage across it drops below the breakdown voltage. This is different from a Zener diode, which will stop conducting once the reverse voltage drops below the breakdown voltage.


Albert Charles Bartlett

Albert Charles Bartlett was an electrical engineer working for the General Electric Company in Wembley. He had some correspondence with Wilhelm Cauer on the subject of filter designs.
He published a treatment of geometrically symmetrical 2-port networks in 1927 and is responsible for Bartlett's bisection theorem which shows that any symmetrical network can be transformed into a symmetrical lattice network.
He also patented the idea of using the method of an active amplifier with "negative resistance" to cancel the inductance of a telephone line.


Tuned amplifier

A tuned amplifier is an electronic amplifier which includes bandpass filtering components within the amplifier circuitry. They are widely used in a variety of wireless applications.
There are several tuning schemes in use,


OpenSim (simulation toolkit)

OpenSim is an open source software system for biomechanical modeling, simulation and analysis. Its purpose is to provide free and widely accessible tools for conducting biomechanics research and motor control science. OpenSim enables a wide range of studies, including analysis of walking dynamics, studies of sports performance, simulations of surgical procedures, analysis of joint loads, design of medical devices, and animation of human and animal movement. The software performs inverse dynamics analysis and forward dynamics simulations. OpenSim is used in hundreds of biomechanics laboratories around the world to study movement and has a community of software developers contributing new features.
OpenSim is one of the flagship applications from Simbios, a NIH Center for Biomedical Computation at Stanford University. Founded in 2004, Simbios is charged with a mandate to provide leading software and computational tools for physics-based modeling and simulation of biological structures. OpenSim was designed to propel biomechanics research by providing a common framework for investigation and a vehicle for exchanging complex musculoskeletal models.
History.
OpenSim 1.0 was released on August 20, 2007 and provided capabilities for viewing musculoskeletal models, importing models developed in SIMM (Musculographics Inc.), editing muscle paths, and generating muscle actuated simulations that track experimental data.
OpenSim 1.1 was released on December 11, 2007, which added new features such as user-specified camera positions for recording movies of simulations, and a perturbation (sensitivity) analysis for inquiry into the function of individual muscles.
OpenSim 2.2.1 was released on April 11, 2011. This software update enhanced the user interface and allowed the user to set bounds on activations of muscles and actuators relating to static optimization not dynamic optimization.
OpenSim 2.4 was released on October 10, 2011. This newest and most recent update includes faster and more robust tools for Inverse Dynamics and Inverse Kinematics, new visualization tools, enhanced access for API users, and many usability improvements.
OpenSim 3.2 was released on March 13, 2014. This update focused on improving the OpenSim scripting interface, accessible through the Graphical User Interface (GUI), Matlab, and now Python. It also added new visualization capabilities and usability improvements in the OpenSim application. Full list of features can be found here.


Calculator-Based Laboratory

The Calculator-Based Laboratory (CBL) is a mobile data collection device manufactured by Texas Instruments in collaboration with Vernier Software & Technology and introduced in 1994. The CBL can be used to collect data either in conjunction with a TI graphing calculator, or on its own.


Dongle

A dongle is a small piece of computer hardware that connects to a port on another device to provide it with additional functionality, or enable a pass-through to such a device that adds functionality.
In computing, the term was initially synonymous with software protection dongles—a form of hardware digital rights management where a piece of software will only operate if a specified dongle—which typically contains a license key or some other cryptographic protection mechanism—is plugged into the computer while it is running.
The term has since been applied to other forms of devices with a similar form factor, such as adapters that convert ports to handle different types of connectors (such as DVI to VGA for displays, USB to serial connection, and in modern computing, USB-C to other types of ports, and Mobile High-Definition Link), USB wireless adapters for standards such as Bluetooth and Wi-Fi (the term "stick" or "key" is more commonly used to describe USB flash drives, as well as modern forms of dongles with slimmer form factors more akin to flash drives), and small form-factor digital media players that plug into HDMI ports.
Etymology.
There are varying accounts on the etymology of the word "dongle"; in a 1999 paper, P. B. Schneck stated that the origin was unclear, but that it was possibly a corruption of the word "dangle" (since these devices "dangle" from a port on a PC).
A 1992 "Byte" magazine advertisement by Rainbow Technologies claimed that dongles were invented by and named after a person named "Don Gall", which spawned an urban legend. Linguist Ben Zimmer noted that the claim was likely a by-product of their "tongue-in-cheek" marketing style, and "was so egregiously false that the company happily owned up to it as a marketing ploy when pressed by Eric S. Raymond, who maintains the Jargon File, an online lexicon of hacker slang."
Examples.
Copy protection.
Security dongles are typically used to help prevent unauthorized use and copying of certain forms of software. Initially using ports such as the serial port or parallel port, most are now in USB format.
On the other hand, some unlicensed game cartridges have a "daisy chain" design that allows licensed games to pass along their authorization, for instance to circumvent the 10NES chip on the Nintendo Entertainment System.
Small peripheral appliances.
In the mid-to-late 2010s, the dongle form factor was extended to digital media players with a small, stick-like form factor — such as Chromecast and Fire TV Stick — that are designed to plug directly into an HDMI port on a television or AV receiver (powered via Micro USB connection to the television itself or an AC adapter), in contrast to a larger set-top box-style device. Single-board computers, such as the Intel Compute Stick, have also been produced in a similar means.


Fujifilm GFX series


Nikon F-mount

The Nikon F-mount is a type of interchangeable lens mount developed by Nikon for its 35mm format single-lens reflex cameras. The F-mount was first introduced on the Nikon F camera in 1958, and features a three-lug bayonet mount with a 44mm throat and a flange to focal plane distance of 46.5mm. The company continues, with the 2020 D6 model, to use variations of the same lens mount specification for its film and digital SLR cameras.
History.
The Nikon F-mount is one of only two SLR lens mounts (the other being the Pentax K-mount) which were not abandoned by their associated manufacturer upon the introduction of autofocus, but rather extended to meet new requirements related to metering, autofocus, and aperture control. The large variety of F-mount compatible lenses makes it the largest system of interchangeable flange-mount photographic lenses in history. Over 400 different Nikkor lenses are compatible with the system (other details can be found at the Nikkor "The Thousand and One Nights" site). The F-mount is also popular in scientific and industrial applications, most notably machine vision. The F-mount has been in production for over five decades, making it the only SLR lens mount which has been produced for over 50 years.
System of lenses.
In addition to Nikon's own range of "Nikkor" lenses, brands of F-mount photographic lenses include Zeiss, Voigtländer, Schneider, Angénieux, Samyang, Sigma, Tokina, Tamron, Hartblei, Kiev-Arsenal, Lensbaby, and Vivitar. F-mount cameras include current models from Nikon, Fujifilm, Sinar, JVC, Kenko and Horseman. Numerous other manufacturers employ the F-mount in non-photographic imaging applications.
Compatibility.
The F-mount has a significant degree of both backward and forward compatibility. Many current autofocus F-mount lenses can be used on the original Nikon F, and the earliest manual-focus F-mount lenses of the 1960s and early 1970s can, with some modification, still be used to their fullest on all professional-class Nikon cameras. Incompatibilities do exist, however, and adventurous F-mount users should consult product documentation in order to avoid problems. For example, many electronic camera bodies cannot meter without a CPU enabled lens; the aperture of G designated lenses cannot be controlled without an electronic camera body; non-AI lenses (manufactured prior to 1977) can cause mechanical damage to later model bodies unless they are modified to meet the AI specification; and AF-P lenses (introduced in 2016) will not focus, even manually, on cameras introduced before roughly 2013. Many manual focus lenses can be converted to allow metering with consumer Nikon bodies by adding a Dandelion chip to the lens.
Image circle.
Most Nikon F-mount lenses cover a minimum of the standard 36×24mm area of 35mm format and the Nikon FX format, while DX designated lenses cover the 24×16mm area of the Nikon DX format, and industrial F-mount lenses have varying coverage. DX lenses may produce vignetting when used on film and FX cameras. However, Nikon lenses designed for film cameras will work on Nikon digital system cameras with the limitations noted above.
Mounting and control rings.
F-mount lenses lock by turning counter-clockwise (when looking at the front of lens) and unlock clockwise. Nearly all F-mount lenses have zoom and focus controls that rotate in the clockwise direction (as viewed from behind the camera) to increase focal length and focus distance respectively. This convention is also used in Pentax K-mount and Sony A-mount lenses but is opposite of the direction normally used by Canon. F-mount lenses also typically have aperture rings that turn clockwise to close.
Nikkor.
Designations.
Nikon has introduced many proprietary designations for F-mount Nikkor lenses, reflecting design variations and developments both in lenses and the F-mount itself. There are also "unofficial" designations used by collectors and dealers to differentiate similar lenses.
Manual-focus lenses.
Perspective control (PC) lenses.
Nikon PC lenses, like other perspective control lenses, offer adjustments that duplicate certain view camera movements. The 28mm and 35mm PC lenses support shifting the lens in relation to the film or sensor plane, while Nikon's 19mm, 24mm, 45mm, and 85mm PC-E lenses also support tilting.
Nikon currently offers four different PC lenses for sale: the four PC-E Nikkors (2008 and 2016), and the 85mm PC-Nikkor (1999). The 45mm and 85mm "Micro" lenses offer close focus (0.5 magnification) for macrophotography. The PC-E lenses (the "E" designates an electromagnetic diaphragm) offer automatic aperture control with all DSLRs with CMOS image sensor except the Nikon D90. With earlier DSLRs and all "analog" film camera models, a PC-E lens operates like a PC lens. The PC Micro-Nikkor 85mm 2.8D lens offers only preset aperture control, actuated mechanically by pressing a plunger.
History.
In July 1962, Nikon released the first interchangeable perspective-control lens available for a single-lens reflex camera, the 35mm 3.5 PC-Nikkor. This was followed in 1968 by a redesigned 35mm 2.8 PC-Nikkor in which the shifting portion of the lens was further from the camera's body, in order to clear the new "Photomic" meters. The last optical redesign of this 35mm lens was released in 1980.
The 35mm PC-Nikkor did not meet the need of photographers for a wider-angle lens, so in July 1975 Nikon released the 28mm 4 PC-Nikkor. In February 1981 Nikon released an improved version of this lens, the 28mm 3.5 PC-Nikkor, with a new optical design. This was the last of the completely manual PC-Nikkors to be offered.
Automatic focus lenses.
Lenses with integrated autofocus motors.
Nikkor lenses designated AF-S, AF-I, or AF-P have integrated autofocus motors, but other manufacturers included in the list do not designate it as clearly. These lenses are needed for autofocus on certain newer low-end Nikon cameras which lack an autofocus motor. These are the Nikon D40, D40X, D60, D3xxx (most recent: D3400), D5xxx (most recent: D5600) and the Nikon 1 series with FT1 adapter.
Other Brands.
Zeiss ZF.
Zeiss ZF series lenses are manual-focus designs Nikon AI-S type aperture indexing. They are manufactured by Cosina to Zeiss specifications.
Four design variations are designated ZF, ZF.2, ZF-I, and ZF-IR.
ZF is the original product line.
ZF.2 lenses are CPU-enabled (similar to Nikon AI-P lenses) offering full metering compatibility with the full range of AF Nikon SLR cameras.
ZF-I lenses add mechanical locks for focus and aperture, and additional environmental sealing, for industrial applications.
ZF-IR lenses are adapted to infrared imaging, with coatings that transmit wavelengths up to 1100 nm, and focus scales marked for infrared.
Zeiss CP.2.
CP.2 lenses are a series of Zeiss "CompactPrime" cinema lenses which present F-mount as one of three mounting options. The lenses cover the 36×24 mm area of the 35mm format or Nikon FX format, and lenses 28 mm and longer share a common T-stop (T/) of 2.1.


Remington Model 7400

The Remington Model 7400 is a series of semi-automatic rifles manufactured by Remington Arms. The Model 7400 was a replacement of the Model 740 rifle which Remington produced from 1952 to 1981. The 7400 model was ultimately replaced by the Model 750 in 2006.
History.
The Model 7400 was introduced in 1981 and was basically the same as the Model Four. Major differences included an checkered pistol-grip and straight comb stock. Initially only offered with a barrel, in 1988 an version was introduced.


Tokarev Sportowy .22 Lr

The Tokarev Sportowy semi-automatic pistol is a Tokarev TT-33 training firearm produced in Poland and used within the former Warsaw Pact countries. While the barrel is sized to receive a .22 caliber projectile, the chamber and magazine are sized to receive 7.62×25mm Tokarev cartridges. Typically, .22 LR cartridges are inserted into a hollow 7.62×25mm Tokarev shaped cartridge, which is then loaded into the firearm.


Donkey Live


Ergodicity

In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies that the average behavior of the system can be deduced from the trajectory of a "typical" point. Equivalently, a sufficiently large collection of random samples from a process can represent the average statistical properties of the entire process. Ergodicity is a property of the system; it is a statement that the system cannot be reduced or factored into smaller components. Ergodic theory is the study of systems possessing ergodicity.
Ergodic systems occur in a broad range of systems in physics and in geometry. This can be roughly understood to be due to a common phenomenon: the motion of particles, that is, geodesics on a hyperbolic manifold are divergent; when that manifold is compact, that is, of finite size, those orbits return to the same general area, eventually filling the entire space.
Ergodic systems capture the common-sense, every-day notions of randomness, such that smoke might come to fill all of a smoke-filled room, or that a block of metal might eventually come to have the same temperature throughout, or that flips of a fair coin may come up heads and tails half the time. A stronger concept than ergodicity is that of mixing, which aims to mathematically describe the common-sense notions of mixing, such as mixing drinks or mixing cooking ingredients.
The proper mathematical formulation of ergodicity is founded on the formal definitions of measure theory and dynamical systems, and rather specifically on the notion of a measure-preserving dynamical system. The origins of ergodicity lie in statistical physics, where Ludwig Boltzmann formulated the ergodic hypothesis.
Informal explanation.
Ergodicity occurs in broad settings in physics and mathematics. All of these settings are unified by a common mathematical description, that of the measure-preserving dynamical system. An informal description of this, and a definition of ergodicity with respect to it, is given immediately below. This is followed by a description of ergodicity in stochastic processes. They are one and the same, despite using dramatically different notation and language. A review of ergodicity in physics, and in geometry follows. In all cases, the notion of ergodicity is "exactly" the same as that for dynamical systems; "there is no difference", except for outlook, notation, style of thinking and the journals where results are published.
Measure-preserving dynamical systems.
The mathematical definition of ergodicity aims to capture ordinary every-day ideas about randomness. This includes ideas about systems that move in such a way as to (eventually) fill up all of space, such as diffusion and Brownian motion, as well as common-sense notions of mixing, such as mixing paints, drinks, cooking ingredients, industrial process mixing, smoke in a smoke-filled room, the dust in Saturn's rings and so on. To provide a solid mathematical footing, descriptions of ergodic systems begin with the definition of a measure-preserving dynamical system. This is written as formula_1
The set formula_2 is understood to be the total space to be filled: the mixing bowl, the smoke-filled room, "etc." The measure formula_3 is understood to define the natural volume of the space formula_2 and of its subspaces. The collection of subspaces is denoted by formula_5, and the size of any given subset formula_6 is formula_7; the size is its volume. Naively, one could imagine formula_5 to be the power set of formula_2; this doesn't quite work, as not all subsets of a space have a volume (famously, the Banach-Tarski paradox). Thus, conventionally, formula_5 consists of the measurable subsets—the subsets that do have a volume. It is always taken to be a Borel set—the collection of subsets that can be constructed by taking intersections, unions and set complements; these can always be taken to be measurable.
The time evolution of the system is described by a map formula_11. Given some subset formula_6, its map formula_13 will in general be a deformed version of formula_14 – it is squashed or stretched, folded or cut into pieces. Mathematical examples include the baker's map and the horseshoe map, both inspired by bread-making. The set formula_13 must have the same volume as formula_14; the squashing/stretching does not alter the volume of the space, only its distribution. Such a system is "measure-preserving" (area-preserving, volume-preserving).
A formal difficulty arises when one tries to reconcile the volume of sets with the need to preserve their size under a map. The problem arises because, in general, several different points in the domain of a function can map to the same point in its range; that is, there may be formula_17 with formula_18. Worse, a single point formula_19 has no size. These difficulties can be avoided by working with the inverse map formula_20; it will map any given subset formula_6 to the parts that were assembled to make it: these parts are formula_22. It has the important property of not losing track of where things came from. More strongly, it has the important property that "any" (measure-preserving) map formula_23 is the inverse of some map formula_24. The proper definition of a volume-preserving map is one for which formula_25 because formula_26 describes all the pieces-parts that formula_14 came from.
One is now interested in studying the time evolution of the system. If a set formula_28 eventually comes to fill all of formula_2 over a long period of time (that is, if formula_30 approaches all of formula_2 for large formula_32), the system is said to be ergodic. If every set formula_14 behaves in this way, the system is a conservative system, placed in contrast to a dissipative system, where some subsets formula_14 wander away, never to be returned to. An example would be water running downhill -- once it's run down, it will never come back up again. The lake that forms at the bottom of this river can, however, become well-mixed. The ergodic decomposition theorem states that every ergodic system can be split into two parts: the conservative part, and the dissipative part.
Mixing is a stronger statement than ergodicity. Mixing asks for this ergodic property to hold between any two sets formula_35, and not just between some set formula_14 and formula_2. That is, given any two sets formula_38, a system is said to be (topologically) mixing if there is an integer formula_39 such that, for all formula_40 and formula_41, one has that formula_42. Here, formula_43 denotes set intersection and formula_44 is the empty set. Other notions of mixing include strong and weak mixing, which describe the notion that the mixed substances intermingle everywhere, in equal proportion. This can be non-trivial, as practical experience of trying to mix sticky, gooey substances shows.
Ergodic processes.
The above discussion appeals to a physical sense of a volume. The volume does not have to literally be some portion of 3D space; it can be some abstract volume. This is generally the case in statistical systems, where the volume (the measure) is given by the probability. The total volume corresponds to probability one. This correspondence works because the axioms of probability theory are identical to those of measure theory; these are the Kolmogorov axioms.
The idea of a volume can be very abstract. Consider, for example, the set of all possible coin-flips: the set of infinite sequences of heads and tails. Assigning the volume of 1 to this space, it is clear that half of all such sequences start with heads, and half start with tails. One can slice up this volume in other ways: one can say "I don't care about the first formula_45 coin-flips; but I want the formula_32'th of them to be heads, and then I don't care about what comes after that". This can be written as the set formula_47 where formula_48 is "don't care" and formula_49 is "heads". The volume of this space is again (obviously!) one-half.
The above is enough to build up a measure-preserving dynamical system, in its entirety. The sets of formula_49 or formula_51 occurring in the formula_32'th place are called cylinder sets. The set of all possible intersections, unions and complements of the cylinder sets then form the Borel set formula_5 defined above. In formal terms, the cylinder sets form the base for a topology on the space formula_2 of all possible infinite-length coin-flips. The measure formula_3 has all of the common-sense properties one might hope for: the measure of a cylinder set with formula_49 in the formula_57'th position, and formula_51 in the formula_59'th position is obviously 1/4, and so on. These common-sense properties persist for set-complement and set-union: everything except for formula_49 and formula_51 in locations formula_57 and formula_59 obviously has the volume of 3/4. All together, these form the axioms of a sigma-additive measure; measure-preserving dynamical systems always use sigma-additive measures. For coin flips, this measure is called the Bernoulli measure.
For the coin-flip process, the time-evolution operator formula_64 is the shift operator that says "throw away the first coin-flip, and keep the rest". Formally, if formula_65 is a sequence of coin-flips, then formula_66. The measure is obviously shift-invariant: as long as we are talking about some set formula_28 where the first coin-flip formula_68 is the "don't care" value, then the volume formula_7 does not change: formula_70. In order to avoid talking about the first coin-flip, it is easier to define formula_71 as inserting a "don't care" value into the first position: formula_72. With this definition, one obviously has that formula_73 with no constraints on formula_14. This is again an example of why formula_71 is used in the formal definitions.
The above development takes a random process, the Bernoulli process, and converts it to a measure-preserving dynamical system formula_76 The same conversion (equivalence, isomorphism) can be applied to any stochastic process. Thus, an informal definition of ergodicity is that a sequence is ergodic if it visits all of formula_2; such sequences are "typical" for the process. Another is that its statistical properties can be deduced from a single, sufficiently long, random sample of the process (thus uniformly sampling all of formula_2), or that any collection of random samples from a process must represent the average statistical properties of the entire process (that is, samples drawn uniformly from formula_2 are representative of formula_2 as a whole.) In the present example, a sequence of coin flips, where half are heads, and half are tails, is a "typical" sequence.
There are several important points to be made about the Bernoulli process. If one writes 0 for tails and 1 for heads, one gets the set of all infinite strings of binary digits. These correspond to the base-two expansion of real numbers. Explicitly, given a sequence formula_65, the corresponding real number is
The statement that the Bernoulli process is ergodic is equivalent to the statement that the real numbers are uniformly distributed. The set of all such strings can be written in a variety of ways: formula_83 This set is the Cantor set, sometimes called the Cantor space to avoid confusion with the Cantor function
In the end, these are all "the same thing".
The Cantor set plays key roles in many branches of mathematics. In recreational mathematics, it underpins the period-doubling fractals; in analysis, it appears in a vast variety of theorems. A key one for stochastic processes is the Wold decomposition, which states that any stationary process can be decomposed into a pair of uncorrelated processes, one deterministic, and the other being a moving average process.
The Ornstein isomorphism theorem states that every stationary stochastic process is equivalent to a Bernoulli scheme (a Bernoulli process with an "N"-sided (and possibly unfair) gaming die). Other results include that every non-dissipative ergodic system is equivalent to the Markov odometer, sometimes called an "adding machine" because it looks like elementary-school addition, that is, taking a base-"N" digit sequence, adding one, and propagating the carry bits. The proof of equivalence is very abstract; understanding the result is not: by adding one at each time step, every possible state of the odometer is visited, until it rolls over, and starts again. Likewise, ergodic systems visit each state, uniformly, moving on to the next, until they have all been visited.
Systems that generate (infinite) sequences of "N" letters are studied by means of symbolic dynamics. Important special cases include subshifts of finite type and sofic systems.
Ergodicity in physics.
Physical systems can be split into three categories: classical mechanics, which describes machines with a finite number of moving parts, quantum mechanics, which describes the structure of atoms, and statistical mechanics, which describes gases, liquids, solids; this includes condensed matter physics. The case of classical mechanics is discussed in the next section, on ergodicity in geometry. As to quantum mechanics, although there is a conception of quantum chaos, there is no clear definition of ergodocity; what this might be is hotly debated. This section reviews ergodicity in statistical mechanics.
The above abstract definition of a volume is required as the appropriate setting for definitions of ergodicity in physics. Consider a container of liquid, or gas, or plasma, or other collection of atoms or particles. Each and every particle formula_85 has a 3D position, and a 3D velocity, and is thus described by six numbers: a point in six-dimensional space formula_86 If there are formula_39 of these particles in the system, a complete description requires formula_88 numbers. Any one system is just a single point in formula_89 The physical system is not all of formula_90, of course; if it's a box of width, height and length formula_91 then a point is in formula_92 Nor can velocities be infinite: they are scaled by some probability measure, for example the Boltzmann–Gibbs measure for a gas. None-the-less, for formula_39 close to Avogadro's number, this is obviously a very large space. This space is called the canonical ensemble.
A physical system is said to be ergodic if any representative point of the system eventually comes to visit the entire volume of the system. For the above example, this implies that any given atom not only visits every part of the box formula_91 with uniform probability, but it does so with every possible velocity, with probability given by the Boltzmann distribution for that velocity (so, uniform with respect to that measure). The ergodic hypothesis states that physical systems actually are ergodic. Multiple time scales are at work: gasses and liquids appear to be ergodic over short time scales. Ergodicity in a solid can be viewed in terms of the vibrational modes or phonons, as obviously the atoms in a solid do not exchange locations. Glasses present a challenge to the ergodic hypothesis; time scales are assumed to be in the millions of years, but results are contentious. Spin glasses present particular difficulties.
Formal mathematical proofs of ergodicity in statistical physics are hard to come by; most high-dimensional many-body systems are assumed to be ergodic, without mathematical proof. Exceptions include the dynamical billiards, which model billiard ball-type collisions of atoms in an ideal gas or plasma. The first hard-sphere ergodicity theorem was for Sinai's billiards, which considers two balls, one of them taken as being stationary, at the origin. As the second ball collides, it moves away; applying periodic boundary conditions, it then returns to collide again. By appeal to homogeneity, this return of the "second" ball can instead be taken to be "just some other atom" that has come into range, and is moving to collide with the atom at the origin (which can be taken to be just "any other atom".) This is one of the few formal proofs that exist; there are no equivalent statements "e.g." for atoms in a liquid, interacting via van der Waals forces, even if it would be common sense to believe that such systems are ergodic (and mixing). More precise physical arguments can be made, though.
Ergodicity in geometry.
Ergodicity is a wide-spread phenomenon in the study of Riemannian manifolds. A quick sequence of examples, from simple to complicated, illustrates this point. All of the systems mentioned below have been proved to be ergodic via rigorous formal proofs. The irrational rotation of a circle is ergodic: the orbit of a point is such that eventually, every other point in the circle is visited. Such rotations are a special case of the interval exchange map. The beta expansions of a number are ergodic: beta expansions of a real number are done not in base-"N", but in base-formula_95 for some formula_96 The reflected version of the beta expansion is tent map; there are a variety of other ergodic maps of the unit interval. Moving to two dimensions, the arithmetic billiards with irrational angles are ergodic. One can also take a flat rectangle, squash it, cut it and reassemble it; this is the previously-mentioned baker's map. Its points can be described by the set of bi-infinite strings in two letters, that is, extending to both the left and right; as such, it looks like two copies of the Bernoulli process. If one deforms sideways during the squashing, one obtains Arnold's cat map. In most ways, the cat map is prototypical of any other similar transformation.
For non-flat surfaces, one has that the geodesic flow of any negatively curved compact Riemann surface is ergodic. A surface is "compact" in the sense that it has finite surface area. The geodesic flow is a generalization of the idea of moving in a "straight line" on a curved surface: such straight lines are geodesics. One of the earliest cases studied is Hadamard's billiards, which describes geodesics on the Bolza surface, topologically equivalent to a donut with two holes. Ergodicity can be demonstrated informally, if one has a sharpie and some reasonable example of a two-holed donut: starting anywhere, in any direction, one attempts to draw a straight line; rulers are useful for this. It doesn't take all that long to discover that one is not coming back to the starting point. (Of course, crooked drawing can also account for this; that's why we have proofs.)
These results extend to higher dimensions. The geodesic flow for negatively curved compact Riemannian manifolds is ergodic. A classic example for this is the Anosov flow, which is the horocycle flow on a hyperbolic manifold. This can be seen to be a kind of Hopf fibration. Such flows commonly occur in classical mechanics, which is the study in physics of finite-dimensional moving machinery, e.g. the double pendulum and so-forth. Classical mechanics is constructed on symplectic manifolds. The flows on such systems can be deconstructed into stable and unstable manifolds; as a general rule, when this is possible, chaotic motion results. That this is generic can be seen by noting that the cotangent bundle of a Riemannian manifold is (always) a symplectic manifold; the geodesic flow is given by a solution to the Hamilton–Jacobi equations for this manifold. In terms of the canonical coordinates formula_97 on the cotangent manifold, the Hamiltonian or energy is given by
with formula_99 the (inverse of the) metric tensor and formula_100 the momentum. The resemblance to the kinetic energy formula_101 of a point particle is hardly accidental; this is the whole point of calling such things "energy". In this sense, chaotic behavior with ergodic orbits is a more-or-less generic phenomenon in large tracts of geometry.
Ergodicity results have been provided in translation surfaces, hyperbolic groups and systolic geometry. Techniques include the study of ergodic flows, the Hopf decomposition, and the Ambrose–Kakutani–Krengel–Kubo theorem. An important class of systems are the Axiom A systems.
A number of both classification and "anti-classification" results have been obtained. The Ornstein isomorphism theorem applies here as well; again, it states that most of these systems are isomorphic to some Bernoulli scheme. This rather neatly ties these systems back into the definition of ergodicity given for a stochastic process, in the previous section. The anti-classification results state that there are more than a countably infinite number of inequivalent ergodic measure-preserving dynamical systems. This is perhaps not entirely a surprise, as one can use points in the Cantor set to construct similar-but-different systems. See measure-preserving dynamical system for a brief survey of some of the anti-classification results.
Historical development.
The idea of ergodicity was born in the field of thermodynamics, where it was necessary to relate the individual states of gas molecules to the temperature of a gas as a whole and its time evolution thereof. In order to do this, it was necessary to state what exactly it means for gases to mix well together, so that thermodynamic equilibrium could be defined with mathematical rigor. Once the theory was well developed in physics, it was rapidly formalized and extended, so that ergodic theory has long been an independent area of mathematics in itself. As part of that progression, more than one slightly different definition of ergodicity and multitudes of interpretations of the concept in different fields coexist.
For example, in classical physics the term implies that a system satisfies the ergodic hypothesis of thermodynamics, the relevant state space being position and momentum space. In dynamical systems theory the state space is usually taken to be a more general phase space. On the other hand in coding theory the state space is often discrete in both time and state, with less concomitant structure. In all those fields the ideas of time average and ensemble average can also carry extra baggage as well—as is the case with the many possible thermodynamically relevant partition functions used to define ensemble averages in physics, back again. As such the measure theoretic formalization of the concept also serves as a unifying discipline. In 1913 Michel Plancherel proved the strict impossibility for ergodicity for a purely mechanical system.
Etymology.
The term "ergodic" is commonly thought to derive from the Greek words ("ergon": "work") and ("hodos": "path", "way"), as chosen by Ludwig Boltzmann while he was working on a problem in statistical mechanics. At the same time it is also claimed to be a derivation of "ergomonode", coined by Boltzmann in a relatively obscure paper from 1884. The etymology appears to be contested in other ways as well.
Definition for discrete-time systems.
Formal definition.
Let formula_102 be a measurable space. If formula_64 is a measurable function from formula_2 to itself and formula_3 a probability measure on formula_102 then we say that formula_64 is formula_3-ergodic or formula_3 is an ergodic measure for formula_64 if formula_64 preserves formula_3 and the following condition holds:
In other words there are no formula_64-invariant subsets up to measure 0 (with respect to formula_3). Recall that formula_64 preserving formula_3 (or formula_3 being formula_64-invariant) means that formula_123 for all formula_113 (see also Measure-preserving dynamical system).
Examples.
The simplest example is when formula_2 is a finite set and formula_3 the counting measure. Then a self-map of formula_2 preserves formula_3 if and only if it is a bijection, and it is ergodic if and only if formula_64 has only one orbit (that is, for every formula_130 there exists formula_131 such that formula_132). For example, if formula_133 then the cycle formula_134 is ergodic, but the permutation formula_135 is not (it has the two invariant subsets formula_136 and formula_137).
Equivalent formulations.
The definition given above admits the following immediate reformulations:
Importantly for applications, the condition in the last characterisation can be restricted to square-integrable functions only:
Further examples.
Bernoulli shifts and subshifts.
Let formula_153 be a finite set and formula_154 with formula_3 the product measure (each factor formula_153 being endowed with its counting measure). Then the shift operator formula_64 defined by formula_158 is formula_3-ergodic.
There are many more ergodic measures for the shift map formula_64 on formula_2. Periodic sequences give finitely supported measures. More interestingly, there are infinitely-supported ones which are subshifts of finite type.
Irrational rotations.
Let formula_2 be the unit circle formula_163, with its Lebesgue measure formula_3. For any formula_165 the rotation of formula_2 of angle formula_167 is given by formula_168. If formula_169 then formula_170 is not ergodic for the Lebesgue measure as it has infinitely many finite orbits. On the other hand, if formula_167 is irrational then formula_170 is ergodic.
Arnold's cat map.
Let formula_173 be the 2-torus. Then any element formula_174 defines a self-map of formula_2 since formula_176. When formula_177 one obtains the so-called Arnold's cat map, which is ergodic for the Lebesgue measure on the torus.
Ergodic theorems.
If formula_3 is a probability measure on a space formula_2 which is ergodic for a transformation formula_64 the pointwise ergodic theorem of G. Birkhoff states that for every measurable functions formula_181 and for formula_3-almost every point formula_183 the time average on the orbit of formula_184 converges to the space average of formula_152. Formally this means that
formula_186
The mean ergodic theorem of J. von Neumann is a similar, weaker statement about averaged translates of square-integrable functions.
Related properties.
Dense orbits.
An immediate consequence of the definition of ergodicity is that on a topological space formula_2, and if formula_188 is the σ-algebra of Borel sets, if formula_64 is formula_3-ergodic then formula_3-almost every orbit of formula_64 is dense in the support of formula_3.
This is not an equivalence since for a transformation which is not uniquely ergodic, but for which there is an ergodic measure with full support formula_194, for any other ergodic measure formula_195 the measure formula_196 is not ergodic for formula_64 but its orbits are dense in the support. Explicit examples can be constructed with shift-invariant measures.
Mixing.
A transformation formula_64 of a probability measure space formula_199 is said to be mixing for the measure formula_3 if for any measurable sets formula_201 the following holds:
formula_202
It is immediate that a mixing transformation is also ergodic (taking formula_14 to be a formula_64-stable subset and formula_205 its complement). The converse is not true, for example a rotation with irrational angle on the circle (which is ergodic per the examples above) is not mixing (for a sufficiently small interval its successive images will not intersect itself most of the time). Bernoulli shifts are mixing, and so is Arnold's cat map.
This notion of mixing is sometimes called strong mixing, as opposed to weak mixing which means that
formula_206
Proper ergodicity.
The transformation formula_64 is said to be "properly ergodic" if it does not have an orbit of full measure. In the discrete case this means that the measure formula_3 is not supported on a finite orbit of formula_64.
Definition for continuous-time dynamical systems.
The definition is essentially the same for continuous-time dynamical systems as for a single transformation. Let formula_102 be a measurable space and for each formula_211, then such a system is given by a family formula_212 of measurable functions from formula_2 to itself, so that for any formula_214 the relation formula_215 holds (usually it is also asked that the orbit map from formula_216 is also measurable). If formula_3 is a probability measure on formula_102 then we say that formula_212 is formula_3-ergodic or formula_3 is an ergodic measure for formula_64 if each formula_212 preserves formula_3 and the following condition holds:
Examples.
As in the discrete case the simplest example is that of a transitive action, for instance the action on the circle given by formula_230 is ergodic for Lebesgue measure.
An example with infinitely many orbits is given by the flow along an irrational slope on the torus: let formula_231 and formula_232. Let formula_233; then if formula_234 this is ergodic for the Lebesgue measure.
Ergodic flows.
Further examples of ergodic flows are:
Ergodicity in compact metric spaces.
If formula_2 is a compact metric space it is naturally endowed with the σ-algebra of Borel sets. The additional structure coming from the topology then allows a much more detailed theory for ergodic transformations and measures on formula_2.
Functional analysis interpretation.
A very powerful alternate definition of ergodic measures can be given using the theory of Banach spaces. Radon measures on formula_2 form a Banach space of which the set formula_238 of probability measures on formula_2 is a convex subset. Given a continuous transformation formula_64 of formula_2 the subset formula_242 of formula_64-invariant measures is a closed convex subset, and a measure is ergodic for formula_64 if and only if it is an extreme point of this convex.
Existence of ergodic measures.
In the setting above it follows from the Banach-Alaoglu theorem that there always exists extremal points in formula_242. Hence a transformation of a compact metric space always admits ergodic measures.
Ergodic decomposition.
In general an invariant measure need not be ergodic, but as a consequence of Choquet theory it can always be expressed as the barycenter of a probability measure on the set of ergodic measures. This is referred to as the "ergodic decomposition" of the measure.
Example.
In the case of formula_246 and formula_247 the counting measure is not ergodic. The ergodic measures for formula_64 are the uniform measures formula_249 supported on the subsets formula_136 and formula_251 and every formula_64-invariant probability measure can be written in the form formula_253 for some formula_254. In particular formula_255 is the ergodic decomposition of the counting measure.
Continuous systems.
Everything in this section transfers verbatim to continuous actions of formula_256 or formula_257 on compact metric spaces.
Unique ergodicity.
The transformation formula_64 is said to be uniquely ergodic if there is a unique Borel probability measure formula_3 on formula_2 which is ergodic for formula_64.
In the examples considered above, irrational rotations of the circle are uniquely ergodic; shift maps are not.
Probabilistic interpretation: ergodic processes.
If formula_262 is a discrete-time stochastic process on a space formula_263, it is said to be ergodic if the joint distribution of the variables on formula_264 is invariant under the shift map formula_265. This is a particular case of the notions discussed above.
The simplest case is that of an independent and identically distributed process which corresponds to the shift map described above. Another important case is that of a Markov chain which is discussed in detail below.
A similar interpretation holds for continuous-time stochastic processes though the construction of the measurable structure of the action is more complicated.
Ergodicity of Markov chains.
The dynamical system associated with a Markov chain.
Let formula_153 be a finite set. A Markov chain on formula_153 is defined by a matrix formula_268, where formula_269 is the transition probability from formula_270 to formula_271, so for every formula_272 we have formula_273. A stationary measure for formula_274 is a probability measure formula_275 on formula_153 such that formula_277 ; that is formula_278 for all formula_272.
Using this data we can define a probability measure formula_280 on the set formula_154 with its product σ-algebra by giving the measures of the cylinders as follows:
formula_282
Stationarity of formula_275 then means that the measure formula_280 is invariant under the shift map formula_158.
Criterion for ergodicity.
The measure formula_280 is always ergodic for the shift map if the associated Markov chain is irreducible (any state can be reached with positive probability from any other state in a finite number of steps).
The hypotheses above imply that there is a unique stationary measure for the Markov chain. In terms of the matrix formula_274 a sufficient condition for this is that 1 be a simple eigenvalue of the matrix formula_274 and all other eigenvalues of formula_274 (in formula_290) are of modulus <1.
Note that in probability theory the Markov chain is called ergodic if in addition each state is aperiodic (the times where the return probability is positive are not multiples of a single integer >1). This is not necessary for the invariant measure to be ergodic; hence the notions of "ergodicity" for a Markov chain and the associated shift-invariant measure are different (the one for the chain is strictly stronger).
Moreover the criterion is an "if and only if" if all communicating classes in the chain are recurrent and we consider all stationary measures.
Examples.
Counting measure.
If formula_291 for all formula_292 then the stationary measure is the counting measure, the measure formula_293 is the product of counting measures. The Markov chain is ergodic, so the shift example from above is a special case of the criterion.
Non-ergodic Markov chains.
Markov chains with recurring communicating classes are not irreducible are not ergodic, and this can be seen immediately as follows. If formula_294 are two distinct recurrent communicating classes there are nonzero stationary measures formula_295 supported on formula_296 respectively and the subsets formula_297 and formula_298 are both shift-invariant and of measure 1.2 for the invariant probability measure formula_299. A very simple example of that is the chain on formula_300 given by the matrix formula_301 (both states are stationary).
A periodic chain.
The Markov chain on formula_300 given by the matrix formula_303 is irreducible but periodic. Thus it is not ergodic in the sense of Markov chain though the associated measure formula_3 on formula_305 is ergodic for the shift map. However the shift is not mixing for this measure, as for the sets
formula_306
and
formula_307
we have formula_308 but
formula_309
Generalisations.
Ergodic group actions.
The definition of ergodicity also makes sense for group actions. The classical theory (for invertible transformations) corresponds to actions of formula_310 or formula_256.
Quasi-invariant measures.
For non-abelian groups there might not be invariant measures even on compact metric spaces. However the definition of ergodicity carries over unchanged if one replaces invariant measures by quasi-invariant measures.
Important examples are the action of a semisimple Lie group (or a lattice therein) on its Furstenberg boundary.
Ergodic relations.
A measurable equivalence relation it is said to be ergodic if all saturated subsets are either null or conull.


Mad Catz

Mad Catz Global Limited (formerly Mad Catz Interactive, Inc.) is an American company that provides interactive entertainment products marketed under Mad Catz, GameShark (gaming products) and TRITTON (audio products). Mad Catz developed flight simulation software through its internal ThunderHawk Studios, developed chess hardware and flight simulation under its Saitek brand, published games under its Mad Catz brand, and distributed games and video game products for third-party partners. The company was incorporated in Canada and headquartered in San Diego, California. Mad Catz had offices in North America, Europe and Asia.
History.
1989–2000.
Mad Catz was founded in 1989. It focused on accessories such as control pads, memory cards, connection cables, headphones and other human interface devices for the PC and various video game consoles. Mad Catz also published original software titles, such as "Real World Golf" 1 and 2, "MC Groovz Dance Craze", and "Pump It Up".
Mad Catz Interactive, Inc. was incorporated under the Canada Business Corporations Act on August 25, 1993. The company was acquired in 2000 by the Toronto-based GTR Group, Inc. for a purchase cost of US$33.3 million, along with Games Trader (which collected and sold previously played and republished games) and ZapYou.com (which focused on e-commerce solutions).
2000–2009.
In 2000, Mad Catz released the MC2 Racing Wheel for the Sony PlayStation, which was awarded the Golden Award by "Incite Video Gaming Magazine", and the Mario Andretti Racing Wheel, which GameSpy judged as the best peripheral of the 2000 Electronic Entertainment Expo (E3). The company produced 12 licensed controllers for the launch of the Sega Dreamcast and also released Internet-related accessories such as the Panther DC and keyboard adapter.
In September 2001, GTR Group decided to capitalize on the strength of the Mad Catz brand and changed its corporate name to Mad Catz Interactive, Inc., closing its GamesTrader and ZapYou.com business units. The year 2001 saw the release of Microsoft's Xbox and Nintendo's GameCube and Game Boy Advance. Mad Catz offered products for these launches, such as the Control Pad Pro for GameCube, the Lynx Control Pad and the BeatPad (dancepad) controller for the PlayStation 2, the Game Boy FlipLight, and memory cards for the Xbox.
In January 2003, Mad Catz acquired the GameShark brand, described as "the industry leader in video game enhancement software, [enabling] players to take full advantage of the secret codes, short cuts, hints and cheats incorporated by video game publishers into their game offerings," and associated intellectual properties from InterAct for US$5 million.
In September 2007, Mad Catz acquired UK-based company Joytech for US$3.7 million. In November 2007, Mad Catz further diversified into the videogame accessory market by acquiring European-based manufacturer Saitek for US$30 million.
In 2008, Capcom commissioned Mad Catz to produce a limited number of ArcadeSticks and six-button controllers, branded as “FightSticks” and “FightPads,” to coincide with the launch of their "Street Fighter IV" fighting game. Due to the unanticipated demand for these products, production on these controllers was ramped up from their originally limited numbers. The FightStick remains a mainstay of the Mad Catz product line. In 2008, the FightStick's popularity spurred Mad Catz' involvement in the pro-gaming community, and Mad Catz has a strong presence at shows such as EVO, PAX, and San Diego Comic-Con. In 2009, Mad Catz released licensed controllers for "" and "."
2010–2017.
In 2010, Mad Catz released officially licensed controllers for "Rock Band 3", including a keyboard controller, a microphone controller, drum kits, and two guitar controllers (both the familiar five-button and the Fender Mustang “Pro” 102-button guitar). That same year, Mad Catz acquired TRITTON Technologies, a San Diego-based gaming audio headset firm, in May 2010 for US$1 million at closing. In June 2010, Mad Catz announced an agreement with Major League Gaming (MLG) to produce the MLG Pro Circuit Controller for Xbox 360 and PlayStation 3. Mad Catz sponsored MLG Pro Circuit Competitions.
In 2011, Mad Catz’ Cyborg-branded R.A.T.7 Gaming Mouse was named “Best PC Accessory of 2010” by IGN. The company returned to software distribution by publishing "Jonah Lomu Rugby Challenge", "Damage Inc. Pacific Squadron WWII" (a World War II flight simulation game with a custom joystick), and "Rock Band 3." Mad Catz acquired certain assets of V Max Simulation Corporation, which designs, constructs, integrates and operates flight simulation equipment and develops flight simulation software.
Mad Catz has continued its focus on peripherals and accessories, such as the range of R.A.T. gaming mice and TRITTON-branded gaming headsets. In January 2013, Mad Catz announced its "GameSmart" initiative, a range of products using the Bluetooth Smart stack featuring game and hardware independence, simplified setup, longer battery life and universal compatibility. Announced products include the R.A.T.M Wireless Mobile Gaming Mouse, the F.R.E.Q.M Mobile Stereo Headset, M.O.U.S.9 Wireless Mouse, and the C.T.R.L.R Mobile Gamepad. The "GameSmart" initiative is intended to introduce an industry standardization to the mobile world.
In June 2013, Mad Catz announced the M.O.J.O. Android Micro Console, described as "a supercharged smart phone with no screen that plugs into your flat screen TV to bring the living room experience to mobile gaming." M.O.J.O. is designed to interact seamlessly with Mad Catz's GameSmart controllers, mice, keyboards, headsets, and the rest of its gaming peripherals. In 2013, Mad Catz produced the "Killer Instinct" Arcade FightStick Tournament Edition 2, the first fighting game controller available for the Xbox One, the then-latest Microsoft gaming console, that started shipping in late November 2013.
Mad Catz helped spur interest in the burgeoning field of eSports by sponsoring the inaugural ESL One at Madison Square Garden in October 2014, as well as sponsoring the Capcom Pro Tour featuring "Street Fighter IV."
In April 2015, Mad Catz announced that it would co-publish "Rock Band 4" along with Harmonix, in addition to making the controllers for the game. Under the terms of the agreement, Mad Catz was responsible for worldwide retail sales, promotion, and distribution for the "Rock Band 4" game and its hardware bundles. Harmonix handled "Rock Band 4" digital sales and content. "Rock Band 4" was launched on October 6, 2015.
In August 2015, Mad Catz announced that it will work with Cloud Imperium Games to create licensed simulation products for its space simulation game "Star Citizen". These products, produced under Mad Catz's Saitek brand, were first unveiled at Gamescom 2015.
On February 9, 2016, Mad Catz announced that it would lay off 37 percent of its workforce and stated that ""Rock Band" sell-through was lower than originally forecast resulting in higher inventory balances as well as lower margins due to increased promotional activity with retailers." This followed the resignations of company president and CEO Darren Richardson, senior VP of business affairs Whitney Peterson, and company chairman Thomas Brown the day prior.
On September 15, 2016, Logitech acquired Saitek, Mad Catz' simulation brand specializing in computer joysticks for $13 million.
In March 2017, the New York Stock Exchange reported to the company that it was in the stages of delisting the company from the Exchange due to "abnormally low" stock value, which Mad Catz does not plan to appeal.
Bankruptcy.
Mad Catz ceased operations on March 30, 2017, and filed a voluntary petition for relief under Chapter 7 of U.S. Bankruptcy code to initiate an orderly liquidation of its assets.
Return.
On 4 January 2018, Mad Catz Global Limited (a new company headquartered in Kowloon, Hong Kong), with new ownership of previous Mad Catz trademarks, announced the return of the Mad Catz brand and the launch of a new line of products at CES 2018. Mad Catz Global Limited has continued the brand's development of gaming peripherals such as mice, keyboards, headphones, and controllers.


Zero fret

A zero fret is a fret placed at the headstock end of the neck of a banjo, guitar, mandolin, or bass guitar. It serves one of the functions of a nut: holding the strings the correct distance above the other frets on the instrument's fretboard. A separate nut is still required to establish the correct string spacing when a zero fret is used.
Function.
The zero fret is positioned at the location normally occupied by the nut. On a guitar having a zero fret, the nut is located behind the zero fret and serves solely to keep the strings spaced properly. The strings rest atop the zero fret, which is generally at the same height as all the others. Some people prefer and feel more comfortable with the zero fret slightly taller than the rest of the frets. The zero fret functions as all other frets do.
A nut is used for spacing. When the zero fret is used the nut is cut lower than the zero fret. If you think it out the second, third and fourth frets etc are the same height. So the zero fret can be the height of all the others.
Purpose.
It is claimed that with a zero fret, the sound of an open string more closely approximates the sound of a fretted string as compared to the open string sound on a guitar with no zero fret. Countering this claim are musicians who feel that a bone or even synthetic nut will enhance the overall tone of the instrument regardless of the string being played open or fretted. Since tone is so subjective, the two claims are likely to continue perpetually.
Some manufacturers that frequently use(d) a zero fret are Gretsch, Kay, Selmer, Höfner, Mosrite, Framus, Vox, Vigier, Harley Benton and bass guitar manufacturer MTD. Now very few manufacturers use this design and those who do list it as a feature. Steinberger uses a zero fret with their headless guitars but omit the nut; strings are mounted in place where the head would normally be, so there is no need for the string guides that the nut provides. 2015 model year Gibson guitars incorporate a zero fret in order to accommodate a brass adjustable nut, which the manufacturer claims causes better sustain and intonation. The British acoustic guitar manufacturer Fylde Guitars uses a zero fret as standard.
Drawbacks.
Low string action (distance between string and fret wire) results from a non-elevated zero fret when using the same fingerboard fret wire size. On some guitars it may be necessary to raise the bridge saddle height by a small amount.
Advantages.
A large number of different gauges of strings can be used. Strings reside on top of the zero fret regardless of thickness, and have the same distance travel down to the first fret. If you have cut the grooves into the nut for thick strings, it may be necessary to change the nut out completely in order to go back to lighter gauge strings. On zero fret, this isn't needed.
A conventional nut made of relatively soft plastic or bone will easily clamp the string and make fine tuning difficult. The clamping effect does not only result from too narrow nut slots, but notably more from the impression of the string windings into the nut material at the bottom of the slot. That effect is more prevalent with wire wound nylon strings than with steel strings. Using a zero fret relieves the pressure from the nut material and the nut serves only to center the strings sideways. Tuning is smooth and without sudden movement and intonation jumps. There are only a few manufacturers making metal (bronze)conventional nuts which avoid the string clamping effect.
Elevated and Straight Zero Fret.
When the zero fret is viewed as an extension of the fingerboard by adding one more fret wire before the nut, then the luthier would fit a fret wire of identical dimensions. This will decrease the string action at the lower fret positions. To avoid this effect, luthiers often use a thicker fret wire at the zero fret position. While the straight and level zero fret improves the tuning accuracy along the lower frets (from fret 1 to 3 mainly) the elevated fret keeps the string action higher and helps to avoid string buzz.
Intonation Issues.
The term "guitar intonation" denotes to what precision the ideal equally tempered musical scale can be produced by a guitar, while "piano intonation" refers to the setup procedure of the individual hammers in the piano keyboard. A conventional nut or an elevated zero fret will cause pitch errors on the lower frets due to the increased pressure on the strings. The fret distances would have to be corrected for that increased tension effect, but with guitars that is not common. On the contrary the "level" zero fret will have no pitch errors on the fingered notes. This can be demonstrated by measuring the pitch deviation of each single note on the fingerboard. Players who frequently use the lower frets from 0 to 5 will benefit most from a zero fret. Using mainly the upper range of the fingerboard neutralizes the advantage of the zero fret and conventional nuts are equally suited. Those players usually avoid playing first and second fret positions because of the pitch problems there. For beginners a zero fretted guitar is preferable.
Most guitars nowadays are manufactured with rather high string action and high conventional nuts. The user is expected to adjust nut height to their personal playing style. Unfortunately sales personnel in music stores DO know about that, but lack the skill and often skilled luthiers are not within reach; not to mention the extra time and/or cost to properly set up a guitar in what has become very price competitive, low profit product. Guitars with zero frets would be helpful in that situation.
The photograph shows a high conventional nut on a factory made Japanese concert guitar and may serve to illustrate the necessity for evaluating the advantages and the drawbacks of modern guitar manufacturing. The high string tension makes the instrument almost unplayable for beginner students, especially for children, and additionally serves to spoil the ear training for harmonies and tone intervals due to intonation errors. Luthier intervention is required in such cases.
High string action at the nut is often preferred to avoid string buzz with heavy playing style. The lower fret area is therefore out of tune. A partial solution is to apply nut corrections by reducing the distance from the nut to fret one for a small amount and then re-tune the now shorter string. That procedure lowers the tuning of all the other notes on the complete string and compromises have to be found. It may no longer be feasible to tune the open string to its base tone. Some manufacturers suggest to tune at the third or fifth fret using electronic tuners.


96th Test Wing

The 96th Test Wing is a United States Air Force unit assigned to the Air Force Test Center of Air Force Materiel Command at Eglin Air Force Base, Florida. The wing was activated at Eglin in 1994 as the 96th Air Base Wing, the headquarters for all support units on Eglin, the largest installation in the Air Force. In 2012, it absorbed the mission and resources of the 46th Test Wing and added the mission of testing and evaluating weapons, navigation and guidance systems and command and control systems.
The wing's first predecessor was organized during World War II as the 96th Bombardment Group. After training in the United States, the group flew Boeing B-17 Flying Fortresses from England. The group led the first shuttle mission to Regensburg on 17 August 1943. The group earned two Distinguished Unit Citations for its combat performance. After VE Day, the group returned to the United States and was inactivated. The group was briefly active in the Air Force Reserve from 1947 until 1949.
The 96th Bombardment Wing was activated in 1953 at Altus Air Force Base, Oklahoma and received Boeing B-47 Stratojet bombers the following year as a component of Strategic Air Command's deterrent force. In 1957 the wing moved to Dyess Air Force Base, Texas where it converted to the Boeing B-52 Stratofortress in 1963 and the Rockwell B-1 Lancer in 1985. The wing also operated air refueling aircraft, and during the early 1960s was assigned a squadron of intercontinental ballistic missiles. In 1984, the World War II group was consolidated with the wing. The wing was inactivated in 1993 and its mission, personnel and equipment were transferred to the 7th Bomb Wing, which moved on paper to Dyess when Carswell Air Force Base became a reserve installation.
Mission.
The 96th Test Wing performs developmental test and evaluation for Air Force weapons while also providing support for all other units on Eglin Air Force Base as the installation host wing. Eglin is the Department of Defense's largest Air Force installation. Supported units include three wings, the Armament Systems Directorate, nine operating locations, five detachments and more than 25 associate units.
History.
World War II.
The group was first activated in July 1942 at Salt Lake City Army Air Base, Utah as the 96th Bombardment Group, with the 337th, 338th, 339th and 413th Bombardment Squadrons assigned as its original components. After moving to Gowen Field, Idaho the group received its initial cadre. The group trained at various bases in the northwestern United States.
In November 1942 the group moved to Pocatello Army Air Base, Idaho, where it acted as an Operational Training Unit (OTU). OTUs were oversized parent units that provided cadres to form "satellite groups." In early 1943, the 96th relocated to Pyote Army Air Base, Texas, where it resumed its combat training. In April 1943 the group began its overseas movement. The air echelon ferried its bombers via the North Atlantic Ferry Route, while the ground echelon proceeded to the New York Port of Embarkation and sailed on the for Greenock, Scotland.
The group arrived at RAF Grafton Underwood England in May 1943, for duty with Eighth Air Force. The group was assigned to the 45th Combat Bombardment Wing of the 3d Bombardment Division. The group commenced combat operations on 14 May with an attack on Kortrijk (Courtrai), after an aborted mission the previous day. The 96th moved east at the end of May to RAF Andrews Field. The 96th appears to have only carried out one mission while based at Andrews. On 29 May 1943 they took part in a raid on Rennes naval storage depot from which one B-17 failed to return. However, Eighth Air Force was not pleased with the initial performance of the Martin B-26 Marauder units assigned to it and decided to move them from their bases in north Suffolk to stations nearer the continent. As the first step in this move, the 386th Bombardment Group left its base at RAF Snetterton Heath for RAF Boxted. The 96th took the 386th's place at Snetterton Heath the following day, leaving its previous base available for the 322d Bombardment Group.
As the most conveniently reached station from 3d Air Division Headquarters at Elveden Hall, Snetterton Heath units often led to major operations carrying commanding generals. General Curtis LeMay led the Regensburg shuttle mission to North Africa flying out of this base, and the group received a Distinguished Unit Citation for withstanding severe assaults by enemy fighters. The 96th also led the 3d Division on the Schweinfurt mission of 14 October 1943. In addition, the 96th attacked shipyards, harbors, railway yards, aerodromes, oil refineries, aircraft factories, and other industrial targets in Germany, France, the Netherlands, Belgium, Norway, Poland, Hungary, and Czechoslovakia.
The 96th received another Distinguished Unit Citation for leading the 45th Wing a great distance through heavy clouds and intense flak to raid important aircraft component factories in Poland on 9 April 1944. Other significant targets attacked by the group included airfields at Bordeaux and Augsburg; marshalling yards at Kiel, Hamm, Brunswick, and Gdynia; aircraft factories at Chemnitz, Hanover, and Diósgyőr; oil refineries at Merseburg and Brüx, and chemical works in Wiesbaden, Ludwigshafen, and Neunkirchen.
In addition to its strategic operations, the 96th was occasionally diverted to support ground forces. These missions included bombing coastal defenses, railway bridges, gun emplacements, and field batteries in the battle area prior to and during D-Day in June 1944. It attacked enemy positions to support the breakthrough at Saint-Lô in July 1944 and aided the campaign in France in August by striking roads and road junctions and by dropping supplies to the Maquis. In the early months of 1945, the group struck lines of communications supplying German armies on the western front.
After V-E Day, the 96th flew food to the Netherlands and transported redeploying personnel to French Morocco, Northern Ireland, France, and Germany. the group was programmed to move to Germany for occupation duty. However, plans were revised in September 1945. In November 1945 its aircraft were flown back to the United States or transferred to other units. Two of its squadrons were inactivated in late November and the others in mid-December. The group headquarters' remaining personnel left Snetterton Heath, sailing on the on 12 December and arriving at Camp Kilmer New Jersey on 20 December 1945, where it was inactivated the following day.
Air Force Reserve.
The 96th Bombardment Group was activated as a reserve unit under Air Defense Command (ADC) on 29 May 1947 at Gunter Field, Alabama. It was initially assigned three of its World War II units, the 337th Bombardment Squadron, located at Gunter, and the 338th and 339th Bombardment Squadrons, stationed at Hawkins Field in Mississippi. In July, it added three additional squadrons, the 413th Bombardment Squadron at Keesler Air Force Base in Mississippi, the 546th Bombardment Squadron at Lovell Field in Tennessee, and the 547th Bombardment Squadron at Smith Reynolds Airport in North Carolina. In October, however, the two squadrons at Jackson were transferred to the 384th Bombardment Group.
At Gunter, the group conducted routine training activities under the supervision of the 476th AAF Base Unit (later the 2586th Air Force Reserve Training Center). President Truman's reduced 1949 defense budget required reductions in the number of units in the Air Force, and the 96th Group and 2586th Center were inactivated in July 1949 and not replaced as flying operations at Gunter ceased.
Strategic Air Command.
The 96th Bombardment Wing was activated in November 1953 at Altus Air Force Base, Oklahoma. Wing headquarters and most of the wing components were not manned until March 1954; those components were controlled by the 96th Air Base Group, whose commander served additional duty as the 96th's wing commander. The wing soon received Boeing KC-97 Stratotankers and began air refueling operations in March 1954.
The wing began training with Boeing B-47E Stratojet swept-wing medium bombers, capable of flying at high subsonic speeds, in April 1955 in support of SAC's global commitments. It deployed to Andersen Air Force Base Guam from January through April 1957. The wing joined the 341st Bombardment Wing at Dyess Air Force Base, Texas on 8 September 1957.
The 4th Strategic Support Squadron, a strategic airlift squadron flying Douglas C-124 Globemaster IIs, moved to Dyess from Ellsworth Air Force Base in 1957. It was assigned to the wing from September 1959 until it was inactivated in March 1961. The wing added intercontinental ballistic missiles to its nuclear strike force when the 578th Strategic Missile Squadron, with Convair SM-65 Atlas missiles joined the wing in July 1961. The first Atlas missile went on alert in April 1962. The wing's Atlases were phased out in March 1965.
Soon after detection of Soviet missiles in Cuba, on 22 October 1962, SAC's B-47s were dispersed. Dispersing bombers carried nuclear weapons in ferrying configuration. On 24 October SAC went to DEFCON 2, placing all its combat aircraft on alert. Most dispersal bases were civilian airfields with Air Force Reserve or Air National Guard units. The B-47s were configured for execution of the Emergency War Order as soon as possible after dispersal. On 15 November, 1/6 of the dispersed B-47s were recalled to their home bases. On 21 November SAC relaxed its alert posture to DEFCON 3. its dispersed B-47s and their supporting tankers were recalled on 24 November. On 27 November SAC returned to normal alert posture.
By the early 1960s, the B-47 was being phased out of SAC's strategic arsenal. In June 1961, the 341st Bombardment Wing had been inactivated and the 96th became the single Stratojet wing at Dyess. In March 1963, two of the wing's bomber squadrons were inactivated, and by December 1963, its remaining squadron had converted to the Boeing B-52 Stratofortress. The 96th received B-52Cs from the 99th Bomb Wing from Walker Air Force Base in Roswell, New Mexico. In 1970 and again 1972–1973, most wing personnel and all of its aircraft and crews deployed to the Pacific in support of the War in Vietnam. During most of this time, the 96BW also supported Operation Chrome Dome missions over the North Pole in the Cold War.
From 1980, the wing's 917th Air Refueling Squadron's Boeing KC-135A Stratotanker aircraft regularly deployed to Europe, Alaska, and the Pacific to support SAC tanker task force requirements.
The 337th Bombardment Squadron became the first squadron to operate the Rockwell B-1B Lancer, after the arrival of the first aircraft in June 1985. In October 1986, B-1Bs assumed SAC Cold War alert duties for the first time. In addition, the 338th Combat Crew Training Squadron received B-1Bs in June 1992.
917th Squadron tankers provided refueling support to units involved in Operation Just Cause, the December 1989 incursion that replaced Manuel Noriega as ruler of Panama. The following August, they ferried personnel and equipment to Andersen Air Force Base, Guam for further movement to Southwest Asia. In December 1990 all remaining tanker aircraft and crews, except those on alert duty, were sent to Incirlik Air Base, Turkey in support of Operation Desert Storm.
Post Cold War.
On 1 September 1991, the wing was redesignated as the 96th Wing and implemented the objective wing concept. It was relieved from assignment to SAC and assigned to Air Combat Command on 1 June 1992. Also on 1 June 1992, the 917th Squadron acquired KC-135Qs in conjunction with the drawdown of tanker operations at Beale Air Force Base, California.
When tanker squadrons were reassigned to Air Mobility Command, the 917th was reassigned to the 43d Operations Group, headquartered at Malmstrom Air Force Base, Montana on 30 September 1993.
On 1 October 1993 the 96th Wing inactivated, replaced by the 7th Wing, which moved without personnel or equipment due to the 1993 Base Realignment and Closure Commission transfer of Carswell Air Force Base, Texas to the U.S. Navy . The B-1Bs of the 337th Squadron were reassigned to the 7th Wing, and the 337th absorbed the B-1s of the inactivating 338th Crew Training Squadron as part of the new wing.
The 96th Air Base Wing stood up as a non-flying organization on 15 March 1994. It assumed the mission of supporting the Air Armament Center and associate units at Eglin Air Force Base, Florida.
Air Force Materiel Command Restructuring.
The wing became the 96th Test Wing, assuming the 46th Test Wing mission, as part of the 2012 Air Force Materiel Command Restructuring. It is the test and evaluation unit for Air Force air-delivered weapons, navigation and guidance systems, command and control systems and Air Force Special Operations Command systems.
The wing performs developmental test and evaluation for a wide variety of customers including: Air Force Systems Program Offices, the Air Force Research Laboratory, Materiel Command's logistics and product centers; major commands; other Department of Defense services and U.S. government agencies (Department of Transportation, NASA, etc.); foreign military sales; and private industry.
Lineage.
Components.
Groups
Squadrons


Debye (crater)

Debye is a lunar impact crater that is located in the northern hemisphere on the Moon's far side, as seen from the Earth. It lies to the south of the crater Chappell, to the southwest of the walled plain Rowland, and to the east of D'Alembert.
The outer rim of this crater has been heavily battered by impacts, and is somewhat distorted from the form of a circle. The northeast rim in particular has become straightened by impact modification, and the crater rim as a whole has a roughly polygonal shape. The southern rim is overlaid by the smaller crater Perkin. The most heavily eroded section of the rim is along the northeast, where a cluster of overlapping craters intrudes into the side. The western rim is notched and indented by several small impacts.
The crater interior is nearly as battered as the outer rim. The outer rampart of Perkin intrudes part way across the southern floor. Parts of remaining interior have been bombarded and churned by impacts, leaving an irregular surface that is nearly as rough as the terrain that surrounds the crater. The most recent of these impacts is a small, cup-shaped crater just to the southwest of the midpoint.
Satellite craters.
By convention these features are identified on lunar maps by placing the letter on the side of the crater midpoint that is closest to Debye.


Clarence Francis

Clarence Francis (December 1, 1888 - December 22, 1985) was a business executive and internationally recognized expert on food.
Early life.
He was born in Staten Island in 1888. Upon graduating from Amherst College in 1910 he went to New York City intending to apply for work at Standard Oil. By mistake, Francis went to the wrong office and soon found himself working for the Corn Products Refining Company, a small food manufacturing firm. This inauspicious beginning led to a lifelong interest in the food industry. By the time Francis retired from the General Foods Corporation in 1954 he had risen to become Chairman of the Board of Directors. After absorbing an initial $17 million loss in the frozen foods sector, Francis was able to turn a profit for General Foods in this very important market.
Career.
After working for Corn Products and Ralston Purina, Francis joined the Postum Company (later General Foods) in 1924. He became executive vice president in 1931, president in 1935,and then chairman of the board in 1943.
In addition to his work in the food industry, Francis also served as a government consultant. By 1933 he was recognized as an expert on food production and distribution. When the National Recovery Administration, one of President Franklin D. Roosevelt’s New Deal agencies, was created to establish codes to regulate business operations, Francis was hired as a consultant to help draft codes for the food industry. The successful conclusion of this task led to further demands for his services. Between 1940 and 1943 he assisted in defense mobilization; from 1943 to 1945 he led a committee that studied the food services at several naval bases in the United States; he studied manpower training and utilization in the U.S. Department of Defense from 1951-52; and he evaluated the Mutual Security Program in 1953.
In 1949, Francis was appointed by Secretary of Interior Julius A. Krug to chair an official United States Citizens Committee for the United Nations Scientific Conference on Conservation and Utilization of Resources, held in August of that year at Lake Success, New York.
Francis’ consultant work required taking extended leaves from his work at the General Foods Corporation. After his retirement in 1954 he was able to devote most of his time to government service. He became a special consultant to President Dwight D. Eisenhower with the responsibility of disposing of the vast agricultural surpluses which the federal government had accumulated. At the same time he served as chairman of the Citizens Committee for the Hoover Report, and was the American representative on the European Productivity Agency.
At the end of the Eisenhower administration Francis left government services and returned to private business. His last major activity was with the Economic Development Council of New York City.
He remained a Director Emeritus of General Foods until his death.


586th Flight Test Squadron

The 586th Flight Test Squadron is part of the 96th Test Wing based at Holloman Air Force Base, New Mexico.
Mission.
The 586th Flight Test Squadron plans, analyzes, coordinates and conducts flight tests of advanced weapons and avionics systems primarily on the White Sands Missile Range.
It provides deployable operational support for test aircraft staging out of Holloman AFB. The squadron flight tests guidance systems, laser systems, air-to-air/air-to-ground systems, long-range and standoff weapons, live warheads, and provides target and photo/safety chase.
The 586th operates three AT-38B aircraft. These aircraft are modified for test, test support, target, and photo/safety chase. Capabilities of the squadron's AT-38B's include: chaff, flares, Global Positioning System (GPS) navigation and precision data recording and telemetry, electronic counter- measures (ECM), towed target, threat and cruise missile simulation, Air Combat Maneuvering Instrumentation pods, and multiple format photographic coverage (including helmet-mounted video cameras). They are equipped with an internal Fighter Instrumentation and Navigation System which relies on inertial navigation and global positioning inputs to develop a reference for time-space-position information. Each aircraft has a 200-ft AGL capability utilizing radar altimeters and moving map displays. For specialized tests, customer provided test equipment may be rack mounted and installed in place of the rear ejection seat or externally in a pod.
Externally, the aircraft has a modified centerline pylon to enable carriage of many types of test and operational stores such as the ALQ-167 Electronic Counter Measures (ECM) pod, which is programmable with a wide variety of electronic jamming techniques as well as an ALE-40 chaff and flare pod. External stores can be provided with AC and DC power. Another test capability under development is a Low Observable Instrumented Tow Target system that will support many different types of tests. Flight cleared pods are available for carriage of additional customer defined stores.
History.
World War II.
Formed under III Bomber Command in early 1943 as a Martin B-26 Marauder medium bomber squadron. Trained for duty in Europe with Ninth Air Force. Engaged in combat beginning in early 1944, attacked tactical targets in France, Low Countries and Germany supporting Allied ground forces advancing after D-Day in Northern France Campaign and the Western Allied invasion of Germany, 1945. Received Douglas A-26 Invaders in April 1945, however did not use in combat. After V-E Day, was part of the Army of Occupation in Germany and inactivated in early 1946.
Test operations.
It has provided operational and maintenance support for all Department of Defense test aircraft staging out of Holloman Air Force Base from, 1982–1990. The squadron flew developmental test and evaluation missions and has supported DOD test flights on White Sands Missile Range since 1991.
Lineage.
586th Bombardment Squadron
586th Test Squadron


Ghostbusters Spooktacular