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Aperture (antenna)

In electromagnetics and antenna theory, the aperture of an antenna is defined as "A surface, near or on an antenna, on which it is convenient to make assumptions regarding the field values for the purpose of computing fields at external points. The aperture is often taken as that portion of a plane surface near the antenna, perpendicular to the direction of maximum radiation, through which the major part of the radiation passes."

Effective area
The effective area of an antenna is defined as "In a given direction, the ratio of the available power at the terminals of a receiving antenna to the power flux density of a plane wave incident on the antenna from that direction, the wave being polarization matched to the antenna." Of particular note in this definition is that both effective area and power flux density are functions of incident angle of a plane wave. Assume a plane wave from a particular direction $$(\theta, \phi)$$, which are the azimuth and elevation angles relative to the array normal, has a power flux density $$\|\vec{S}\|$$; this is the amount of power passing through a unit area normal to the direction of the plane wave of one square meter.

By definition, if an antenna delivers $$P_\text{O}$$ watts to the transmission line connected to its output terminals when irradiated by a uniform field of power density $$|S(\theta, \phi)|$$ watts per square meter, the antenna's effective area $$A_\text{e}$$ for the direction of that plane wave is given by
 * $$A_\text{e}(\theta, \phi) = \frac{P_O}{\|\vec{S}(\theta, \phi)\|}.$$

The power $$P_\text{O}$$ accepted by the antenna (the power at the antenna terminals) is less than the power $$P_\text{R}$$ received by an antenna by the radiation efficiency $$\eta$$ of the antenna. $$P_\text{R}$$ is equal to the power density of the electromagnetic energy $$|S(\theta, \phi)| = |\vec{S} \cdot \hat{a}|$$, where $$\hat{a}$$ is the unit vector normal to the array aperture, multiplied by the physical aperture area $$A$$. The incoming radiation is assumed to have the same polarization as the antenna. Therefore,
 * $$P_\text{O} = \eta P_\text{R} = \eta A |\vec{S} \cdot \hat{a}| = \eta A \|\vec{S}(\theta, \phi)\| \cos\theta \cos\phi,$$

and
 * $$A_\text{e}(\theta, \phi) = \eta A \cos\theta \cos\phi.$$

The effective area of an antenna or aperture is based upon a receiving antenna. However, due to reciprocity, an antenna's directivity in receiving and transmitting are identical, so the power transmitted by an antenna in different directions (the radiation pattern) is also proportional to the effective area $$A_e$$. When no direction is specified, $$A_e$$ is understood to refer to its maximal value.

Effective length
Most antenna designs are not defined by a physical area but consist of wires or thin rods; then the effective aperture bears no clear relation to the size or area of the antenna. An alternate measure of antenna response that has a greater relationship to the physical length of such antennas is effective length $$l_\text{eff}$$ measured in metres, which is defined for a receiving antenna as

 * $$l_\text{eff} = V_0 / E_\text{s},$$

where
 * $$V_0$$ is the open-circuit voltage appearing across the antenna's terminals,
 * $$E_s$$ is the electric field strength of the radio signal, in volts per metre, at the antenna.

The longer the effective length, the greater is the voltage appearing at its terminals. However, the actual power implied by that voltage depends on the antenna's feedpoint impedance, so this cannot be directly related to antenna gain, which is a measure of received power (but does not directly specify voltage or current). For instance, a half-wave dipole has a much longer effective length than a short dipole. However the effective area of the short dipole is almost as great as it is for the half-wave antenna, since (ideally), given an ideal impedance-matching network, it can receive almost as much power from that wave. Note that for a given antenna feedpoint impedance, an antenna's gain or $$A_\text{eff}$$ increases according to the square of $$l_\text{eff}$$, so that the effective length for an antenna relative to different wave directions follows the square root of the gain in those directions. But since changing the physical size of an antenna inevitably changes the impedance (often by a great factor), the effective length is not by itself a useful figure of merit for describing an antenna's peak directivity and is more of theoretical importance. In practice, the effective length of a particular antenna is often combined with its impedance and loss to become the realized effective length.

Aperture efficiency
In general, the aperture of an antenna cannot be directly inferred from its physical size. However so-called aperture antennas such as parabolic dishes and horn antennas, have a large (relative to the wavelength) physical area $$A_\text{phys}$$ which is opaque to such radiation, essentially casting a shadow from a plane wave and thus removing an amount of power $$A_\text{phys} S$$ from the original beam. That power removed from the plane wave can be actually received by the antenna (converted into electrical power), reflected or otherwise scattered, or absorbed (converted to heat). In this case the effective aperture $$A_e$$ is always less than (or equal to) the area of the antenna's physical aperture $$A_\text{phys}$$, as it accounts only for the portion of that wave actually received as electrical power. An aperture antenna's aperture efficiency $$e_\text{a}$$ is defined as the ratio of these two areas:
 * $$e_\text{a} = \frac{A_e}{A_\text{phys}}.$$

The aperture efficiency is a dimensionless parameter between 0 and 1 that measures how close the antenna comes to using all the radio wave power intersecting its physical aperture. If the aperture efficiency were 100%, then all the wave's power falling on its physical aperture would be converted to electrical power delivered to the load attached to its output terminals, so these two areas would be equal: $$A_\text{e} = A_\text{phys}$$. But due to nonuniform illumination by a parabolic dish's feed, as well as other scattering or loss mechanisms, this is not achieved in practice. Since a parabolic antenna's cost and wind load increase with the physical aperture size, there may be a strong motivation to reduce these (while achieving a specified antenna gain) by maximizing the aperture efficiency. Aperture efficiencies of typical aperture antennas vary from 0.35 to well over 0.70.

Note that when one simply speaks of an antenna's "efficiency", what is most often meant is the radiation efficiency, a measure which applies to all antennas (not just aperture antennas) and accounts only for the gain reduction due to losses. Outside of aperture antennas, most antennas consist of thin wires or rods with a small physical cross-sectional area (generally much smaller than $$A_\text{e}$$) for which "aperture efficiency" is not even defined.

Aperture and gain
The directivity of an antenna, its ability to direct radio waves preferentially in one direction or receive preferentially from a given direction, is expressed by a parameter $$G$$ called antenna gain. This is most commonly defined as the ratio of the power $$P_\text{o}$$ received by that antenna from waves in a given direction to the power $$P_\text{iso}$$ that would be received by an ideal isotropic antenna, that is, a hypothetical antenna that receives power equally well from all directions. It can be seen that (for antennas at a given frequency) gain is also equal to the ratio of the apertures of these antennas:
 * $$G = \frac{P_\text{o}}{P_\text{iso}} = \frac{A_\text{e}}{A_\text{iso}}.$$

As shown below, the aperture of a lossless isotropic antenna, which by this definition has unity gain, is
 * $$A_\text{iso} = \frac{\lambda^2}{4\pi},$$

where $$\lambda$$ is the wavelength of the radio waves. Thus
 * $$G = \frac{A_\text{e}}{A_\text{iso}} = \frac{4\pi A_\text{e}}{\lambda^2}.$$

So antennas with large effective apertures are considered high-gain antennas (or beam antennas), which have relatively small angular beam widths. As receiving antennas, they are much more sensitive to radio waves coming from a preferred direction compared to waves coming from other directions (which would be considered interference). As transmitting antennas, most of their power is radiated in a particular direction at the expense of other directions. Although antenna gain and effective aperture are functions of direction, when no direction is specified, these are understood to refer to their maximal values, that is, in the direction(s) of the antenna's intended use (also referred to as the antenna's main lobe or boresight).

Friis transmission formula
The fraction of the power delivered to a transmitting antenna that is received by a receiving antenna is proportional to the product of the apertures of both the antennas and inversely proportional to the squared values of the distance between the antennas and the wavelength. This is given by a form of the Friis transmission formula:
 * $$\frac{P_\text{r}}{P_\text{t}} = \frac{A_\text{r} A_\text{t}}{d^2 \lambda^2},$$

where
 * $$P_\text{t}$$ is the power fed into the transmitting antenna input terminals,
 * $$P_\text{r}$$ is the power available at receiving antenna output terminals,
 * $$A_\text{r}$$ is the effective area of the receiving antenna,
 * $$A_\text{t}$$ is the effective area of the transmitting antenna,
 * $$d$$ is the distance between antennas (the formula is only valid for $$d$$ large enough to ensure a plane wave front at the receive antenna, sufficiently approximated by $$d \gtrsim 2a^2/\lambda$$, where $$a$$ is the largest linear dimension of either of the antennas),
 * $$\lambda$$ is the wavelength of the radio frequency.

Derivation of antenna aperture from thermodynamic considerations
The aperture of an isotropic antenna, the basis of the definition of gain above, can be derived on the basis of consistency with thermodynamics. Suppose that an ideal isotropic antenna A with a driving-point impedance of R sits within a closed system CA in thermodynamic equilibrium at temperature T. We connect the antenna terminals to a resistor also of resistance R inside a second closed system CR, also at temperature T. In between may be inserted an arbitrary lossless electronic filter Fν passing only some frequency components.

Each cavity is in thermal equilibrium and thus filled with black-body radiation due to temperature T. The resistor, due to that temperature, will generate Johnson–Nyquist noise with an open-circuit voltage whose mean-squared spectral density is given by
 * $$\overline{v_n^2} = 4 k_\text{B} T R \, \eta(f),$$

where $$\eta(f)$$ is a quantum-mechanical factor applying to frequency f; at normal temperatures and electronic frequencies $$\eta(f) = 1$$, but in general is given by
 * $$\eta(f) = \frac{hf/k_\text{B} T}{e^{hf/k_\text{B} T} - 1}.$$

The amount of power supplied by an electrical source of impedance R into a matched load (that is, something with an impedance of R, such as the antenna in CA) whose rms open-circuit voltage is vrms is given by
 * $$P = \frac{\text{v}_\text{rms}^2}{4\text{R}}.$$

The mean-squared voltage $$\overline{v_n^2} = \text{v}_\text{rms}^2$$ can be found by integrating the above equation for the spectral density of mean-squared noise voltage over frequencies passed by the filter Fν. For simplicity, let us just consider Fν as a narrowband filter of bandwidth B1 around central frequency f1, in which case that integral simplifies as follows:
 * $$P_R = \frac{\int_0^\infty 4 k_\text{B} T R \, \eta(f) \, F_\nu(f) \, df}{4\text{R}}$$
 * $$\qquad = \frac{4 k_\text{B} T R \, \eta(f_1) \, B_1}{4\text{R}} = k_\text{B} T \, \eta(f_1) \, B_1.$$

This power due to Johnson noise from the resistor is received by the antenna, which radiates it into the closed system CA.

The same antenna, being bathed in black-body radiation of temperature T, receives a spectral radiance (power per unit area per unit frequency per unit solid angle) given by Planck's law:
 * $$\text{P}_{f,A,\Omega}(f) = \frac{2hf^3}{c^2} \frac{1}{e^{hf / k_\text{B} T} - 1}

= \frac{2f^2}{c^2} \, k_\text{B} T \, \eta(f),$$ using the notation $$\eta(f)$$ defined above.

However, that radiation is unpolarized, whereas the antenna is only sensitive to one polarization, reducing it by a factor of 2. To find the total power from black-body radiation accepted by the antenna, we must integrate that quantity times the assumed cross-sectional area Aeff of the antenna over all solid angles Ω and over all frequencies f:
 * $$P_A = \int_0^\infty \int_{4\pi} \, \frac{P_{f,A,\Omega}(f)}{2} A_\text{eff}(\Omega, f) \, F_\nu(f) \, d\Omega \, df.$$

Since we have assumed an isotropic radiator, Aeff is independent of angle, so the integration over solid angles is trivial, introducing a factor of 4π. And again we can take the simple case of a narrowband electronic filter function Fν which only passes power of bandwidth B1 around frequency f1. The double integral then simplifies to
 * $$P_A = 2\pi P_{f,A,\Omega}(f) A_\text{eff} \, B_1

= \frac{4\pi \, k_\text{B} T \, \eta(f_1)}{\lambda_1^2} A_\text{eff} B_1,$$ where $$\lambda_1 = c/f_1$$ is the free-space wavelength corresponding to the frequency f1.

Since each system is in thermodynamic equilibrium at the same temperature, we expect no net transfer of power between the cavities. Otherwise one cavity would heat up and the other would cool down in violation of the second law of thermodynamics. Therefore, the power flows in both directions must be equal:
 * $$P_A = P_R.$$

We can then solve for Aeff, the cross-sectional area intercepted by the isotropic antenna:
 * $$\frac{4 \pi \, k_\text{B} T \, \eta(f_1)}{\lambda_1^2} A_\text{eff} B_1

= k_\text{B} T \, \eta(f_1) \, B_1,$$
 * $$A_\text{eff} = \frac{\lambda_1^2}{4\pi}.$$

We thus find that for a hypothetical isotropic antenna, thermodynamics demands that the effective cross-section of the receiving antenna to have an area of λ2/4π. This result could be further generalized if we allow the integral over frequency to be more general. Then we find that Aeff for the same antenna must vary with frequency according to that same formula, using λ = c/f. Moreover, the integral over solid angle can be generalized for an antenna that is not isotropic (that is, any real antenna). Since the angle of arriving electromagnetic radiation only enters into Aeff in the above integral, we arrive at the simple but powerful result that the average of the effective cross-section Aeff over all angles at wavelength λ must also be given by

Although the above is sufficient proof, we can note that the condition of the antenna's impedance being R, the same as the resistor, can also be relaxed. In principle, any antenna impedance (that isn't totally reactive) can be impedance-matched to the resistor R by inserting a suitable (lossless) matching network. Since that network is lossless, the powers PA and PR will still flow in opposite directions, even though the voltage and currents seen at the antenna and resistor's terminals will differ. The spectral density of the power flow in either direction will still be given by $$k_\text{B} T \, \eta(f)$$, and in fact this is the very thermal-noise power spectral density associated with one electromagnetic mode, be it in free-space or transmitted electrically. Since there is only a single connection to the resistor, the resistor itself represents a single mode. And an antenna, also having a single electrical connection, couples to one mode of the electromagnetic field according to its average effective cross-section of $$\lambda_1^2/(4\pi)$$.

title: Aperture (antenna)


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Diane Finley

Diane Finley (born October 3, 1957) is a former Canadian politician. From 2006 through 2015, she served in the cabinet of Prime Minister Stephen Harper. Her ministerial portfolios included Minister of Human Resources and Skills Development, Minister of Public Works and Government Services, Minister Responsible for Canada Mortgage and Housing Corporation, or CMHC, and Minister of Citizenship and Immigration. She was a member of the House of Commons of Canada, representing the riding of Haldimand—Norfolk for the Conservative Party from 2004 to 2021. In August 2020, she announced that she would not be running in the 2021 Canadian federal election. She resigned from office on May 11, 2021.

Personal life
Diane Finley was raised in Port Dover and Charlotteville in Norfolk County, Ontario, and has a Bachelor of Arts degree and a Masters in Business Administration from the University of Western Ontario. After graduation, she became the administrator of Western's French Immersion School. She then worked in a series of private and public sector jobs, in a variety of fields. She was named "one of Canada's future leaders" by the Governor-General's Study Conference 2000. As a child, she participated in Girl Guides of Canada youth programs.

Her husband was Doug Finley, a Senator from Ontario and former campaign manager and director of political operations for the Conservative Party of Canada. The pair met while Diane was working as a summer hire at Rolls-Royce where Doug was an executive.

In 2006, Finley announced that she has Graves' disease, a non-life-threatening thyroid condition that causes increased sensitivity to bright lights and forces her to wear tinted glasses.

Political career
Long involved in politics of the Conservative Party and its predecessors, Finley first ran for public office in the 2004 federal election in the riding of Haldimand—Norfolk. She defeated Bob Speller, a Liberal cabinet minister, by 1,645 votes. After being re-elected in the 2006 election, she was appointed to the cabinet as Minister of Human Resources and Skills Development. An Order in Council transferred authority for Social Development Canada to her as well, and accordingly, she served under the style Minister of Human Resources and Social Development. She was shuffled from the Human Resources and Skills Development Canada to the Citizenship and Immigration Canada portfolio on January 4, 2007. The following year, Finley allegedly received threats from sex industry officials in relation to her support of Bill C-17, which sought to allow immigration officers to deny temporary visas to prospective strippers if they were suspected to be sex trafficking victims. Tim Lambrinos of the Adult Entertainment Association of Canada (AEAC) said that "it's not plausible" that any of the AEAC strip clubs were responsible for the threats.

Finley was re-elected in the 2008 election and resumed her former post as Minister of Human Resources and Skills Development. She was re-elected in the 2011 election with 50.9% of the vote in her riding.

In March 2015, a parliamentary ethics report on an affair linked to disgraced prime ministerial aide Nigel Wright found that Finley had breached conflict of interest rules in her capacity as minister by diverting funding to a favoured project in Markham whose promoter had close ties to the Conservative Party.

After winning her seat once more in the 2015 federal election, Finley announced that she would be running for the position of interim leader of the Conservative Party, after Stephen Harper's resignation from the post. She was passed over for interim leader and subsequently placed in Rona Ambrose's shadow cabinet.

She was re-elected in the 2019 federal election. On August 23, 2020, she announced that she would not seek re-election in the 2021 federal election. She resigned from the House of Commons on May 11, 2021. Leslyn Lewis was the Conservative candidate elected to succeed Finley the 2021 federal election.

Political views
Finley has promoted increased private-sector involvement in health services. She was the founder of Canada's largest publicly funded ambulance service company, Canadian Medical Response, and has been active in the Canadian Council for Public-Private Partnerships.

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

title: Diane Finley


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Liebig (crater)

Liebig is a lunar impact crater that is located to the south-southeast of the crater Mersenius, and west of the Mare Humorum in the southwest part of the Moon. To the southwest of Liebig is the slightly smaller crater de Gasparis.

The rim of Liebig has a low inner wall, with an interior floor that is nearly featureless except for a few tiny craterlets. It is nearly, but not quite circular, with outward bulges to the west and southeast. The small, cup-shaped crater Liebig A lies along the inner wall and part of the interior floor.

The western edge of the Mare Humorum is a fault line designated the Rupes Liebig, named after this crater. This fault extends for about 180 km along the shore. To the southwest, in the section of lava-flooded surface between Liebig and de Gasparis, is a system of rilles named the Rimae de Gasparis. These cover an area with a diameter of 130 km.

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 Liebig.

title: Liebig (crater)


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The New Scooby-Doo Movies

The New Scooby-Doo Movies is an American animated mystery comedy television series produced by Hanna-Barbera for CBS. It is the second television series in the Scooby-Doo franchise, and follows the first incarnation, Scooby-Doo, Where Are You! It premiered on September 9, 1972, and ended on October 27, 1973, running for two seasons on CBS as the only hour-long Scooby-Doo series. Twenty-four episodes were produced, sixteen for the 1972–73 season and eight more for the 1973–74 season.

Aside from doubling the length of each episode, The New Scooby-Doo Movies differed from its predecessor in the addition of a rotating special guest star slot; each episode featured real world celebrities or well-known animated characters joining the Mystery, Inc. gang in solving mysteries. This concept was later revisited with a similar animated series titled Scooby-Doo and Guess Who?, which premiered in 2019.

The New Scooby-Doo Movies was the last incarnation of Scooby-Doo airing on CBS, and also the franchise's final time to feature Nicole Jaffe as the regular voice of Velma Dinkley, due to her marriage and retirement from acting.

Overview
Many of the guest stars who appeared in The New Scooby-Doo Movies were living celebrities who provided their own voices (Don Knotts, Jerry Reed, Cass Elliot, Jonathan Winters, Sandy Duncan, Tim Conway, Dick Van Dyke, Don Adams, Davy Jones and Sonny & Cher, among others); some episodes featured celebrities who were retired or deceased, whose voicing was done by imitators (The Three Stooges and Laurel and Hardy), and the rest were crossovers with present or future Hanna-Barbera characters.

The characters from other Hanna-Barbera shows Harlem Globetrotters, Josie and the Pussycats, Jeannie, and Speed Buggy all appeared on the show during or after their own shows' original runs; The Addams Family and Batman and Robin (the first time the latter duo appeared in a Hanna-Barbera cartoon ) both appeared on the show a year before they were incorporated into Hanna-Barbera shows of their own, The Addams Family and Super Friends, respectively.

After The New Scooby-Doo Movies ended its original network run in August 1974, repeats of Scooby-Doo, Where Are You! aired on CBS for the next two years. No new Scooby-Doo cartoons would be produced until the show moved to ABC in September 1976 on the highly publicized The Scooby-Doo/Dynomutt Hour. When the various Scooby-Doo series entered syndication in 1980, each New Movies episode was halved and run as two half-hour parts. The USA Network Cartoon Express began running the New Movies in their original format beginning in September 1990; they were rerun on Sunday mornings until August 1992.

In 1994, The New Scooby-Doo Movies began appearing on three Turner Broadcasting networks: TNT, Cartoon Network and Boomerang. Like many animated series created by Hanna-Barbera in the 1970s, the show contained a laugh track created by the studio. The first season of the series was animated at Hanna-Barbera's main studio in Los Angeles, while the second season was animated at their newly formed studio, Hanna-Barbera Pty, Ltd. in Australia.

Main

 * Don Messick – Scooby-Doo, Ngogi the Orangutan (in "Ghastly Ghost Town"), Sheriff (in "Ghastly Ghost Town"), Red-Haired Rock Festival Businessman (in "The Frickert Fracas"), Captain Moody (in "Guess Who's Knott Coming to Dinner"), Guard (in "A Good Medium is Rare"), Drawbridge Operator (in "A Good Medium is Rare"), Mr. Van Henstone (in "Sandy Duncan's Jekyll and Hyde"), Slats (in "Sandy Duncan's Jekyll and Hyde"), Matilda/Matt Hildago (in "The Secret of Shark Island"), Mr. Sawyer (in "The Ghost of the Red Baron"), Andrew Terra (in "The Ghost of the Red Baron"), Swampy Pete/Pete Dugan (in "The Ghostly Creep from the Deep"), Haunted Horseman/Duke of Strathmore (in "The Haunted Horseman of Hagglethorn Hill"), Cyrus Wheedly (in "The Haunted Horseman of Hagglethorn Hill"), Sergeant (in "The Phantom of the Country Music Hall"), Professor Flakey (in "The Caped Crusader Caper"), Ghost of Captain Scavenger/Captain Cannaby/Jack Canna (in "The Haunted Showboat"), Prince Abin (in "Mystery in Persia"), Abdullah (in "Mystery in Persia"), Clem Duncan (in "The Weird Winds of Winona"), Mr. Franklin (in "The Haunted Candy Factory"), Ghostly Strongman/Masked Marvel (in "The Haunted Carnival")
 * Casey Kasem – Norville "Shaggy" Rogers, Robin (in "The Dynamic Scooby-Doo Affair" and "The Caped Crusader Caper"), Ling-Poo (in "Wednesday Is Missing"), Brown-Haired Rock Festival Businessman (in "The Frickert Fracas"), Ghost of Captain Moody (in "Guess Who's Knott Coming to Dinner"), Valet (in "A Good Medium is Rare"), Police Officer (in "Sandy Duncan's Jekyll and Hyde"), Car Radio Announcer (in "The Phantom of the Country Music Hall"), Alexander Cabot III (in "The Haunted Showboat"), Ghost of Injun Joe/Jack Canna's Partner (in "The Haunted Showboat"), Sheriff (in "The Weird Winds of Winona")
 * Frank Welker – Fred Jones, Cleopatra (in "Wednesday is Missing"), Male Housekeeper (in "Wednesday is Missing"), Vulture (in "Wednesday is Missing"), Sheriff Dandy Griffith (in "The Spooky Fog of Juneberry"), Possessed Viking Mannequin/Ben Bing (in "The Phantom of the Country Music Hall"), Hot Dog Vendor (in "The Mystery of Haunted Island"), Lance Goodwin (in "The Haunted Showboat"), P.J. Peabody (in "The Weird Winds of Winona")
 * Heather North – Daphne Blake, Shirley Dennison (in "Sandy Duncan's Jekyll and Hyde"), Gypsy Machine (in "The Haunted Carnival")
 * Nicole Jaffe – Velma Dinkley

Special guest stars

 * Don Adams – Himself (in "The Exterminator")
 * Eddie "Rochester" Anderson - Bobby Joe Mason (in "The Ghostly Creep from the Deep", "The Loch Ness Mess", and "The Mystery of Haunted Island")
 * John Astin – Gomez Addams (in "Wednesday Is Missing")
 * Joe Besser – Babu (in "Mystery in Persia")
 * Daws Butler – Larry and Curly Joe (in "Ghastly Ghost Town" and "The Ghost of the Red Baron")
 * Ted Cassidy – Lurch (in "Wednesday Is Missing")
 * Sonny & Cher – Themselves (in "The Secret of Shark Island")
 * Tim Conway – Himself (in "The Spirit Spooked Sports Show"), Mr. Griffith (in "The Spirit Spooked Sports Show"), Mr. Griffith's Twin Brother (in "The Spirit Spooked Sports Show")
 * Jackie Coogan – Uncle Fester (in "Wednesday Is Missing")
 * Scatman Crothers - Meadowlark Lemon (in "The Ghostly Creep from the Deep", "The Loch Ness Mess", and "The Mystery of Haunted Island")
 * Phyllis Diller – Herself (in "A Good Medium Is Rare")
 * Robert DoQui - Pablo Robertson (in "The Ghostly Creep from the Deep", "The Loch Ness Mess", and "The Mystery of Haunted Island")
 * Sandy Duncan – Herself (in "Sandy Duncan's Jekyll and Hyde")
 * Dick Van Dyke – Himself (in "The Haunted Carnival")
 * Cass Elliot – Herself (in "The Haunted Candy Factory")
 * Richard Elkins - J.C. "Gip" Gipson (in "The Ghostly Creep from the Deep", "The Loch Ness Mess", and "The Mystery of Haunted Island")
 * Jodie Foster – Pugsley Addams (in "Wednesday Is Missing")
 * Stu Gilliam - Curly Neal (in "The Ghostly Creep from the Deep", "The Loch Ness Mess", and "The Mystery of Haunted Island")
 * Larry Harmon – Stan Laurel (in "The Ghost of Bigfoot"), Ghost of Bigfoot (in "The Ghost of Bigfoot")
 * Pat Harrington – Moe (in "Ghastly Ghost Town" and "The Ghost of the Red Baron")
 * Cindy Henderson – Wednesday Addams (in "Wednesday Is Missing")
 * Don Knotts – Himself (in "Guess Who's Knott Coming to Dinner?" and "The Spooky Fog of Juneberry")
 * Carolyn Jones – Morticia Addams (in "Wednesday Is Missing")
 * Davy Jones – Himself (in "The Haunted Horseman of Hagglethorn Hall")
 * Jim MacGeorge - Oliver Hardy (in "The Ghost of Bigfoot")
 * Julie McWhirter - Jeannie
 * Jerry Reed – Himself (in "The Phantom of the Country Music Hall")
 * Olan Soule – Batman (in "The Dynamic Scooby-Doo Affair" and "The Caped Crusader Caper")
 * Janet Waldo – Granny Frump (in "Wednesday Is Missing"), Female Housekeeper (in "Wednesday Is Missing"), Josie (in "The Haunted Showboat")
 * Johnny Williams - Geese Ausby (in "The Ghostly Creep from the Deep", "The Loch Ness Mess", and "The Mystery of Haunted Island")
 * Jonathan Winters – Himself (in "The Frickert Fracas"), Maude Frickert (in "The Frickert Fracas")

Additional voices

 * Sherry Alberoni - Alexander Cabot (in "The Haunted Showboat")
 * Michael Bell - Scorpions Coach (in "The Mystery of Haunted Island"), Jesse Finster (in "The Spirit Spooked Sports Show"), Mark (in "The Weird Winds of Winona")
 * Mel Blanc - Scorpions Trainer (in "The Mystery of Haunted Island"), Speed Buggy (in "The Weird Winds of Winona")
 * Henry Corden - Geronimo/Rhino (in "Ghastly Ghost Town"), Tyronosaurus Rex (in "Ghastly Ghost Town"), Jonathan Crabtree's Chauffeur (in "The Ghost of Bigfoot"), Redcoat Ghost (in "The Loch Ness Mess")
 * Jerry Dexter - Alan (in The Haunted Showboat")
 * Jamie Farr
 * June Foray - Mrs. Baker (in "The Dynamic Scooby-Doo Affair")
 * Joan Gerber - Madame Zokar (in "A Good Medium is Rare")
 * Arlene Golonka - Debbie (in "The Weird Winds of Winona")
 * Florence Halop
 * Mark Hamill - Corey Anders (in "Mystery in Persia")
 * Bob Hastings - Henry Glopp (in "Mystery in Persia")
 * Ann Jillian
 * Jackie Joseph - Melody (in "The Haunted Showboat")
 * Ted Knight - Gunslinger/Amos Crunch (in "Ghastly Ghost Town"), Penguin (in "The Dynamic Scooby-Doo Affair" and "The Caped Crusader Caper"), Ghost of Paul Revere (in "The Loch Ness Mess")
 * Phil Luther Jr. - Tinker (in "The Weird Winds of Winona")
 * Alan Oppenheimer
 * Barbara Pariot - Valerie (in "The Haunted Showboat")
 * Cindy Putman - Pizza Parlor Waitress (in "The Phantom of the Country Music Hall")
 * Mike Road - Minuteman Ghost (in "The Loch Ness Mess")
 * John Stephenson - Cousin Itt (in "Wednesday is Missing"), Vernon Crow (in "The Frickert Fracas"), Scarecrow/Simon Shakey (in "The Frickert Fracas"), Alberto (in "A Good Medium is Rare"), Lucas (in "A Good Medium is Rare"), Mysto (in "A Good Medium is Rare"), Duke Jason (in "Sandy Duncan's Jekyll and Hyde"), Mr. Thayer (in "Sandy Duncan's Jekyll and Hyde"), Mr. Hyde/Zalia Z. Fairchild (in "Sandy Duncan's Jekyll and Hyde"), Milo Meekly (in "The Secret of Shark Island"), Hans Edelweiss (in "The Ghost of Bigfoot"), Ghost of the Red Baron/Mr. Siegfried (in "The Ghost of the Red Baron"), Farmer (in "The Ghost of the Red Baron"), Creech (in "The Haunted Horseman of Hagglethorn Hill"), Morgan (in "The Loch Ness Mess"), Selby (in "The Loch Ness Mess"), Winslow (in "The Loch Ness Mess"), Scorpions Owner (in "The Mystery of Haunted Island"), Johnny Briggs (in "The Haunted Showboat"), Jay Teller (in "The Spirit Spooked Sports Show"), Spirit of Fireball McPhan (in "The Spirit Spooked Sports Show"), Lorne Chumley (in "The Exterminator"), Otto (in "The Exterminator"), Green Glob #1/Mr. Crink (in "The Haunted Candy Factory"), Green Glob #2/Sterling Smith (in "The Haunted Candy Factory"), Detective Storm (in "The Haunted Candy Factory")
 * Larry Storch - Joker (in "The Dynamic Scooby-Doo Affair" and "The Caped Crusader Caper")
 * Vincent VanPatten - Possessed Davy Crockett Mannequin/Bertha/Bank Robber (in "The Phantom of the Country Music Hall")
 * Jonathan Walmsley
 * Lennie Weinrib - Robot Gunslinger (in "Ghastly Ghost Town"), Gene Haultry (in "The Spooky Fog of Juneberry"), Redbeard's Ghost (in "The Ghostly Creep from the Deep"), Ghost Pirate #2 (in "The Ghostly Creep from the Deep"), Sir Sedgewick (in "The Haunted Horseman of Hagglethorn Hill"), Andrew Jackson Hotel Desk Clerk (in "The Phantom of the Country Music Hall"), Uncle Nathaniel (in "The Loch Ness Mess")

Season 1

 * Producers/directors: William Hanna and Joseph Barbera
 * Story: Jameson Brewer, Tom Dagenais, Ruth Flippen, Fred Freiberger, Willie Gilbert, Bill Lutz, Larry Markes, Norman Maurer, Jack Mendelsohn, Ray Parker, Gene Thompson, Paul West, Harry Winkler

Season 2

 * Producers/directors: William Hanna and Joseph Barbera
 * Story: Jack Mendelsohn, Tom Dagenais, Norman Maurer, Larz Bourne, Woody Kling, Sid Morse

U.S. sets
Upon attempting to release a complete DVD set of the 24-episode series in 2005, Warner Home Video was unable to negotiate agreements with several of the episodes' guest stars to have those episodes included. As a result, the DVD was released under the title The Best of the New Scooby-Doo Movies, and featured only 15 episodes culled from both seasons. The opening titles on this release were edited to remove the images of the Addams Family, Batman and Robin, the Harlem Globetrotters, the Three Stooges, and Laurel and Hardy.

On April 4, 2019, Warner Bros. announced plans to release eight more episodes, both as part of a package with the 15 previously released episodes and as a standalone release. This release was planned for the 50th anniversary of Scooby-Doo. No explanation for the previous appearances' rights issues was provided.

The only episode that has not been released or announced for release on DVD is "Wednesday Is Missing", which features the Addams Family.

Season set

Volume sets

Other releases

The two episodes featuring Batman and Robin and two of the three episodes featuring the Harlem Globetrotters were also included in two separate releases: Scooby-Doo Meets Batman and Scooby-Doo Meets the Harlem Globetrotters.

The Addams Family episode "Wednesday is Missing" was released to VHS in Australia under the title "Scooby-Doo Meets the Addams Family" and was also released in the U.K. on VHS, along with the Three Stooges episode "Ghastly Ghost Town" under the title "Scooby-Doo Meets the Three Stooges". "The Secret of Shark Island" (featuring Sonny and Cher) episode was released in the U.S. and the U.K. on VHS on a video called Hanna-Barbera Presents: The Best of Scooby-Doo.

title: The New Scooby-Doo Movies


text:
Rosalind Howard, Countess of Carlisle

Rosalind Frances Howard, Countess of Carlisle (née Stanley; 20 February 1845 – 12 August 1921), known as The Radical Countess, was a promoter of women's political rights and temperance movement activist.

Family
The Countess of Carlisle was born probably at Alderley Park, Chelford in Cheshire. She was the tenth and last child of the Whig politician Edward Stanley, 2nd Baron Stanley of Alderley, and the women's education campaigner Henrietta Stanley, Baroness Stanley of Alderley. She was educated at home by private tutors. The Stanley family was exceptionally diverse in terms of religious convictions: Lord and Lady Stanley were high church Anglicans, their eldest son Henry was a Muslim, their third daughter Maude was a low church Anglican, their youngest son Algernon became a Roman Catholic bishop, their penultimate daughter Kate leaned towards atheism, while Rosalind herself identified as an agnostic.

Marriage
On 4 October 1864, she married the painter George Howard, who became an active Liberal MP from 1879. She took part in election campaigns of her husband and father-in-law Charles by canvassing, but refrained from speaking publicly, which was considered improper for a woman. In sharp contrast to her moderate husband, however, Howard soon joined the radical left, denouncing William Ewart Gladstone's occupation of Egypt and campaigning for women's suffrage. She once responded to criticism of herself by saying: "Fanatics have done a lot of the world's work, and I don't mind being classed with the fanatics."

In its early days, the marriage was close and filled with romance. George showered Rosalind with love letters and nude sketches, but the couple gradually drifted apart. They shared a dislike for alcohol, but little else; when the Liberal Party split on the issue of Irish home rule, which Rosalind supported, George decided to side with his cousin, the Duke of Devonshire, and the Liberal Unionist Party. Due to their personal and political disagreements, the Howards spent most of their married life separated, with Rosalind preferring to stay at their country houses, Castle Howard and her favourite home, Naworth Castle.

Views and causes
Despite being plagued by poor health, Howard made use of her organisational skills. She joined Liberal Party women's associations and the temperance movement, involved herself in the management of the extensive family estates and took part in local government. She took the temperance pledge in 1881 and started requiring teetotalism from her tenants and closing down public houses on her estates the next year. Howard gained further credit in 1889 when her husband succeeded his uncle William as 9th Earl of Carlisle, thereby also inheriting the family fortune, and she became known as Countess of Carlisle. In 1891, a United Kingdom Alliance official convinced Lady Carlisle to speak on the subject of temperance at a drawing-room meeting of women. She soon became a successful platform speaker and vice-president of the United Kingdom Alliance, as well as president of the North of England Temperance League in 1892.

In 1890, Lady Carlisle became a member of the Women's Liberal Federation and persuaded the organisation to support extending the suffrage to all women, but denounced Pankhurst suffragettes' violent methods. She presided over the federation from 1894 until 1902 and again from 1906 until 1915. She was elected president of the British Women's Temperance Association in 1903 and president of the World's Woman's Christian Temperance Association in 1906, retaining both offices until her death. Lady Carlisle disagreed with the policy of her predecessor, Lady Henry Somerset, and Thomas Palmer Whittaker, who, among other things, advocated compensating licence holders who lost their livelihoods due to temperance.

The Countess of Carlisle allied herself with a small group of Liberal MPs, including her son Geoffrey, her son-in-law Charles Henry Roberts, her secretary Leifchild Leif-Jones and her neighbour Sir Wilfrid Lawson. The Good Templars supported her policies, but she refused invitations to join the mostly working-class and lower middle-class organisation.

When Lady Carlisle's daughter, Lady Dorothy Georgiana Howard, was attending Girton College, her closest college friends included archaeologist Gisela Richter and future candidate for Roman Catholic Sainthood Anna Abrikosova. During vacations, both were honored guests of Lady Carlisle at Castle Howard and Castle Naworth.

Although she had opposed the South African War, Lady Carlisle firmly supported British resistance to the Germans in the First World War. The temperance movement and the Liberal Party had divided by then, leaving her without significant political influence. She supported H. H. Asquith despite his unwillingness to promote prohibition and opposed David Lloyd George's proposal to nationalise the drink trade during wartime. Though she worked hard to improve the working-class people's living conditions, she was an élitist who resented their role in democracy.

Death and legacy
By the time Lord Carlisle died in 1911, Lady Carlisle's autocracy had estranged her from most of her children and friends. She strongly disapproved of her daughters' flirtatiousness and bitterly argued with her eldest son Charles, a Tory politician. For several years, Lady Carlisle refused to speak to her daughter Dorothy due to her marriage to the brewer Francis Henley (afterwards Baron Henley). Lady Henley later claimed that her mother was privately a tyrant, despite appearing at her best in public.

Her husband left most of the family property to her for life and instructed her to divide it among their children upon her death. Rosalind, Countess of Carlisle, died on 12 August 1921 at her home in Kensington Palace Gardens, having survived all her sons except for Geoffrey, and was cremated at Golders Green Crematorium four days later. Her ashes were interred alongside her husband's at Lanercost Priory on 18 August, with Geoffrey as the chief mourner. The surviving children found her last will and testament to be unfair and agreed to redivide the inheritance. Her daughter Cecilia succeeded her as president of the British Women's Temperance Association.

In popular culture
Lady Carlisle served as a model for Lady Britomart in George Bernard Shaw's play Major Barbara.

Issue
Rosalind and George Howard had 11 children:

 * Charles James Stanley Howard, 10th Earl of Carlisle (8 March 1867 - 20 January 1912).
 * Hon. Hubert George Lyulph Howard (3 April 1871 - September 1898), killed at the Battle of Omdurman.
 * Hon. Christopher Edward Howard (2 June 1873 - 1 September 1896).
 * Hon. Oliver Howard (b. 14 March 1875) married Muriel Stephenson on 17 March 1900 and had issue
 * Hon. Geoffrey William Algernon Howard (12 February 1877 - 20 June 1935).
 * Hon. Michael Francis Stafford Howard (23 January 1880 - 9 October 1917) married Nora Hensman and had issue
 * Lady Mary Henrietta Howard (d. 2 September 1956) married Gilbert Murray on 30 November 1889 and had issue
 * Lady Cecilia Howard (d. 6 May 1947) married Charles Henry Roberts (d. 25 July 1959) on 7 April 1891 and had issue
 * Lady Dorothy Howard (d. 14 September 1968) married Francis Robert Eden (1877–1962) on 14 October 1913 and had issue
 * Lady Elizabeth Dacre Howard (born and died in 1883).
 * Lady Aurea Howard (b. 1884) married Denyss Chamberlaine Wace in 1923 and Major Thomas MacLeod OBE in 1928.

title: Rosalind Howard, Countess of Carlisle


text:
Palmieri (crater)

Palmieri is a lunar impact crater that lies to the southwest of the Mare Humorum, in the southwestern quadrant of the Moon's near side. It lies due south of the crater Liebig and east-northeast of the larger Fourier.

This crater formation has an eroded outer rim that has a gap along the southeastern side and a pair of narrow gaps along the northern rim. The remains of a small, flooded crater form the northern edge of the gap along the southeast. Just to the southwest of Palmieri is the small crater Palmieri E.

The interior floor of Palmieri has been flooded by lava, leaving a dark interior surface that has a similar albedo to the lunar mare to the northeast. This floor is joined with a long expanse of lava surface to the north of Palmieri that proceeds northwards until it reaches the area between Liebig and de Gasparis. The floor is marked by a system of rilles named the Rimae Palmieri that continue to the north and southeast beyond the crater perimeter. These narrow rifts extend for about 150 kilometers.

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 Palmieri.

title: Palmieri (crater)


text:
Hecato of Rhodes

Hecato or Hecaton of Rhodes (Ἑκάτων; fl. c. 100 BC) was a Greek Stoic philosopher.

He was a native of Rhodes, and a disciple of Panaetius, but nothing else is known of his life. It is clear that he was eminent amongst the Stoics of the period. He was a voluminous writer, but nothing remains. Diogenes Laërtius mentions six treatises written by Hecato: In addition Cicero writes that Hecato wrote a work on On Duties, (De Officiis) dedicated to Quintus Tubero. Hecato is also frequently mentioned by Seneca in his treatise De Beneficiis. Seneca also quotes Hecato in his Epistulae morales ad Lucilium;
 * Περὶ ἀγαθῶν – On Goods, in at least nineteen books.
 * Περὶ ἀρετῶν – On Virtues.
 * Περὶ παθῶν – On Passions.
 * Περὶ τελῶν – On Ends.
 * Περὶ παραδόξων – On Paradoxes, in at least thirteen books.
 * Χρεῖαι – Maxims.

Cease to hope, and you will cease to fear. (Epistle V)

What progress, you ask, have I made? I have begun to be a friend to myself.(Epistle VI)

I can show you a philtre, compounded without drugs, herbs, or any witch's incantation: 'If you want to be loved, love.' (Epistle IX)

According to Diogenes, Hecato divided the virtues into two kinds, those founded on scientific intellectual principles (i.e. wisdom and justice), and those with no such basis (e.g., temperance and the resultant health and vigor). Like the earlier Stoics, Cleanthes and Chrysippus, Hecato also held that virtue may be taught.

Cicero shows that he was much interested in casuistical questions, as, for example, whether a good man who received a coin he knew to be bad was justified in passing it on to another. On the whole, he is inclined to regard self-interest as the best criterion. This he modifies by explaining that self-interest is based on the relationships of life; a man needs money for the sake of his children, his friends and the state whose general prosperity depends on the wealth of its citizens: It is a wise man's duty to take care of his private interests, at the same time doing nothing contrary to the civil customs, laws, and institutions. But that depends on our purpose in seeking prosperity; for we do not aim to be rich for ourselves alone but for our children, relatives, friends, and, above all, for our country. For the private fortunes of individuals are the wealth of the state.

title: Hecato of Rhodes


text:
Root test

In mathematics, the root test is a criterion for the convergence (a convergence test) of an infinite series. It depends on the quantity
 * $$\limsup_{n\rightarrow\infty}\sqrt[n]{|a_n|},$$

where $$a_n$$ are the terms of the series, and states that the series converges absolutely if this quantity is less than one, but diverges if it is greater than one. It is particularly useful in connection with power series.

Root test explanation
The root test was developed first by Augustin-Louis Cauchy who published it in his textbook Cours d'analyse (1821). Thus, it is sometimes known as the Cauchy root test or Cauchy's radical test. For a series

 * $$\sum_{n=1}^\infty a_n$$

the root test uses the number

 * $$C = \limsup_{n\rightarrow\infty}\sqrt[n]{|a_n|},$$

where "lim sup" denotes the limit superior, possibly +∞. Note that if

 * $$\lim_{n\rightarrow\infty}\sqrt[n]{|a_n|},$$

converges then it equals C and may be used in the root test instead.

The root test states that:
 * if C < 1 then the series converges absolutely,
 * if C > 1 then the series diverges,
 * if C = 1 and the limit approaches strictly from above then the series diverges,
 * otherwise the test is inconclusive (the series may diverge, converge absolutely or converge conditionally).

There are some series for which C = 1 and the series converges, e.g. $$\textstyle \sum 1/{n^2}$$, and there are others for which C = 1 and the series diverges, e.g. $$\textstyle\sum 1/n$$.

Application to power series
This test can be used with a power series

 * $$f(z) = \sum_{n=0}^\infty c_n (z-p)^n$$

where the coefficients cn, and the center p are complex numbers and the argument z is a complex variable.

The terms of this series would then be given by an = cn(z − p)n. One then applies the root test to the an as above. Note that sometimes a series like this is called a power series "around p", because the radius of convergence is the radius R of the largest interval or disc centred at p such that the series will converge for all points z strictly in the interior (convergence on the boundary of the interval or disc generally has to be checked separately).

A corollary of the root test applied to a power series is the Cauchy–Hadamard theorem: the radius of convergence is exactly $$1/\limsup_{n \rightarrow \infty}{\sqrt[n]{|c_n|}},$$ taking care that we really mean ∞ if the denominator is 0.

Proof
The proof of the convergence of a series Σan is an application of the comparison test.

If for all n ≥ N (N some fixed natural number) we have $$\sqrt[n]{|a_n|} \le k < 1$$, then $$|a_n| \le k^n < 1$$. Since the geometric series $$\sum_{n=N}^\infty k^n$$ converges so does $$\sum_{n=N}^\infty |a_n|$$ by the comparison test. Hence Σan converges absolutely.

If $$\sqrt[n]{|a_n|} > 1$$ for infinitely many n, then an fails to converge to 0, hence the series is divergent.

Proof of corollary: For a power series Σan = Σcn(z − p)n, we see by the above that the series converges if there exists an N such that for all n ≥ N we have

 * $$\sqrt[n]{|a_n|} = \sqrt[n]{|c_n(z - p)^n|} < 1,$$

equivalent to

 * $$\sqrt[n]{|c_n|}\cdot|z - p| < 1$$

for all n ≥ N, which implies that in order for the series to converge we must have $$|z - p| < 1/\sqrt[n]{|c_n|}$$ for all sufficiently large n. This is equivalent to saying

 * $$|z - p| < 1/\limsup_{n \rightarrow \infty}{\sqrt[n]{|c_n|}},$$

so $$R \le 1/\limsup_{n \rightarrow \infty}{\sqrt[n]{|c_n|}}.$$ Now the only other place where convergence is possible is when

 * $$\sqrt[n]{|a_n|} = \sqrt[n]{|c_n(z - p)^n|} = 1,$$

(since points > 1 will diverge) and this will not change the radius of convergence since these are just the points lying on the boundary of the interval or disc, so

 * $$R = 1/\limsup_{n \rightarrow \infty}{\sqrt[n]{|c_n|}}.$$

Examples
Example 1:
 * $$ \sum_{i=1}^\infty \frac{2^i}{i^9} $$

Applying the root test and using the fact that $$ \lim_{n \to \infty} n^{1/n}=1,$$
 * $$ C = \lim_{n \to \infty}\sqrt[n]{\left|\frac{2^n}{n^9}\right|}= \lim_{n \to \infty}\frac{ \sqrt[n]{2^n} } { \sqrt[n]{n^9} } = \lim_{n \to \infty}\frac{ 2 }  {(n^{1/n})^9 }  = 2 $$

Since $$ C=2>1,$$ the series diverges.

Example 2:
 * $$1 + 1 + 0.5 + 0.5 + 0.25 + 0.25 + 0.125 + 0.125 + ...  $$

The root test shows convergence because
 * $$r=\limsup_{n\to\infty}\sqrt[2n]{|a_{2n}|} = \limsup_{n\to\infty}\sqrt[2n]{|.5^n|}=\sqrt{.5}<1.$$

This example shows how the root test is stronger than the ratio test. The ratio test is inconclusive for this series if $$n$$ is odd so $$a_n=a_{n+1} = .5^n$$ (though not if $$n$$ is even), because
 * $$r=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| = \lim_{n\to\infty}\left|\frac{ 2 \cdot.5^{n}}{2 \cdot.5^{n}}\right| =1. $$

Root tests hierarchy
Root tests hierarchy is built similarly to the  ratio tests  hierarchy (see Section 4.1 of  ratio test, and more specifically Subsection 4.1.4 there).

For a series $$\sum_{n=1}^\infty a_n$$ with positive terms we have the following tests for convergence/divergence.

Let $$K\geq1$$ be an integer, and let $$\ln_{(K)}(x)$$ denote the $$K$$th iterate of natural logarithm, i.e. $$\ln_{(1)}(x)=\ln (x)$$ and for any $$2\leq k\leq K$$, $$\ln_{(k)}(x)=\ln_{(k-1)}(\ln (x))$$.

Suppose that $$\sqrt[-n]{a_n}$$, when $$n$$ is large, can be presented in the form

 * $$\sqrt[-n]{a_n}=1+\frac{1}{n}+\frac{1}{n}\sum_{i=1}^{K-1}\frac{1}{\prod_{k=1}^i\ln_{(k)}(n)}+\frac{\rho_n}{n\prod_{k=1}^K\ln_{(k)}(n)}.$$

(The empty sum is assumed to be 0.)

 * The series converges, if $$\liminf_{n\to\infty}\rho_n>1$$
 * The series diverges, if $$\limsup_{n\to\infty}\rho_n<1$$
 * Otherwise, the test is inconclusive.

Proof
Since $$\sqrt[-n]{a_n}=\mathrm{e}^{-\frac{1}{n}\ln a_n}$$, then we have

 * $$\mathrm{e}^{-\frac{1}{n}\ln a_n}=1+\frac{1}{n}+\frac{1}{n}\sum_{i=1}^{K-1}\frac{1}{\prod_{k=1}^i\ln_{(k)}(n)}+\frac{\rho_n}{n\prod_{k=1}^K\ln_{(k)}(n)}.$$

From this,

 * $$ \ln a_n=-n\ln\left(1+\frac{1}{n}+\frac{1}{n}\sum_{i=1}^{K-1}\frac{1}{\prod_{k=1}^i\ln_{(k)}(n)}+\frac{\rho_n}{n\prod_{k=1}^K\ln_{(k)}(n)}\right).$$

From Taylor's expansion applied to the right-hand side, we obtain:

 * $$ \ln a_n=-1-\sum_{i=1}^{K-1}\frac{1}{\prod_{k=1}^i\ln_{(k)}(n)}-\frac{\rho_n}{\prod_{k=1}^K\ln_{(k)}(n)}+O\left(\frac{1}{n}\right).$$

Hence,

 * $$a_n=\begin{cases}\mathrm{e}^{-1+O(1/n)}\frac{1}{(n\prod_{k=1}^{K-2}\ln_{(k)}n)\ln^{\rho_n}_{(K-1)}n}, &K\geq2,\\

\mathrm{e}^{-1+O(1/n)}\frac{1}{n^{\rho_n}}, &K=1. \end{cases} $$ (The empty product is set to 1.)

The final result follows from the integral test for convergence.

title: Root test


text:
Talk:Advanced modular arithmetic theory

Intro
This article in its current form is dog crap, and needs to be re-written. It is meant to be provide an advanced expostion of modular arithmetic that goes beyond the scope of the more basic article modular arithmetic. (User:Linas forgot to sign)

Agreed--this has almost nothing "advanced" in it. Most of this ought to be/is in the basic article. I would vote that instead of having this article, we link to more advanced articles from the basic article. --Superninja 06:39, 10 Mar 2005 (UTC)
 * Theorems about primality tests
 * Structure of the multiplicative group of integers modulo n
 * Lagrange's theorem about polynomials mod primes (at most deg(p) roots)
 * Quadratic residue
 * Quadratic reciprocity
 * Primitive roots

 * Most of the stuff in this article is indeed in the basic article. The stuff which is not, is on purpose. I, and other people, think that the extra stuff in here does not fit well with what is already in modular arithmetic (it is poorly written, and jumps a bit in the level of knowledge required).

 * But all of this is debatable. I would be very happy to discuss this with you.

 * I agree with you that this article is worthless. If it stays longer that way, I could even submit it for deletion or redirect it.

 * If you have time, could you please improve on this article? (I don't have the necessary background). Linas who contributed the material in here (second half) seems to not care much about it, except for doing the cut & paste thing (if you check the history). Oleg Alexandrov 16:06, 10 Mar 2005 (UTC)

 * Oleg, you made a promise on this, which you now seem to be backing out of. The agreement as I understood it was that almost all of the content would be cut out of the modular arithmetic article, including all the stuff about rings, music, art, chinese remainder theorem, etc. and moved to this article. I thought the whole point of these edits was to direct people like Superninja to edit this article, and not the modular arithmetic article, which would be kept spare and simple.  So what happened to this promise?

 * The problems we discussed about modular arithmetic was its idiosyncratic, non-standard point-of-view-ish treatment of the topic, which would be cured by editing everything out of it, and directing readers to this article the longer, more complex, "in-depth" article on the topic.

 * Yet above, you are calling this article "worthless" and a candidate for deletion ... can you make up your mind? And if you do delete this, could you maybe copy my edits back into the modular arithmetic article?  Otherwise, what was the whole point of this strange game?  linas 05:08, 21 Mar 2005 (UTC)

 * My language was too strong. Sorry.

 * I did not see our agreement that way. I imagined that after you copy the stuff over here, you will rewrite it into a totally new article relying on modular arithmetic (without repeating it), and from a higher math perspective. I thought you will not stop with a cut & paste. Let us now see if Superninja plans to do anything about this. Oleg Alexandrov 05:19, 21 Mar 2005 (UTC)

 * Well, wouldn't it be easier to put the sections on number theory, group theory and algorithms back into the modular arithmetic article? Ring theory is an advanced topic, and yet you didn't cut that section out. Why? Why not preface the ring theory by first introducing Euclid's algorithm?  This would make the modular arithmetic article easier to read, easier to understand, easier to grasp. Ditto for number theory, that should also come before ring theory. You felt it necessary to chop out Eucilid's algorithm and yet you kept the section on rings.  Why?  Surely Euclid's algorithm is far more deserving of inclusion in modular arithmetic, than a link to rings and ideals?   Why keep  modular arithmetic opaque and lop-sided and ugly-looking in this way?  Why not make it smooth and even and balanced? linas 15:22, 21 Mar 2005 (UTC)

 * I can answer all these questions. But one should start by asking if you really want to get into this, you mentioned earlier you don't have time for this. Oleg Alexandrov 15:46, 21 Mar 2005 (UTC)

Question
Linas wrote in the article:

 * The first encyclopedia entry on modular arithmetic was written by Euclid, in Book 7 of Euclid's Elements.

Now, was it encyclopedia what Euclid wrote? The Euclid's Elements page says it was a mathematical treatise, aka, an Abramowitz and Stegun before its time. Oleg Alexandrov 22:11, 5 Feb 2005 (UTC)

 * Yes well, encyclopedia is an 18th century Enlightnment concept, so, by definition, anything written before that time cannot technically be called an encyclopedia. On the other hand, the Elements, along with Aristotle's works and Pliny's History, come close to what in modern times would be called an "encyclopedia".  So I think the usage is entirely approriate. A&S is an excellent reference work, but its neither encylopeadic nor is it a treatise. linas 00:17, 6 Feb 2005 (UTC)

 * Now we are getting into fine points about what an encyclopedia is, and whether Euclid wrote about modular arithmetic or ordinary arithmetic. Thus, never mind. Could you check the talk page of Fixed point theorems in infinite-dimensional spaces? I have a question there about the paragaph you inserted. Thanks. Oleg Alexandrov 02:16, 6 Feb 2005 (UTC)

Recommendation
Redirect and merge this page with Modular arithmetic. Then in that article when concepts that are meant for more advanced students, put in links like the one I just did for modular arithmetic and have then link to specific concepts. This would be easily done via a list of them in the main article.
 * Some stuff in this article is of rather bad quality. By merging back this and the other planned things, we would get a huge article ranging from the fine details implementation of modulo function in computing to very abstract algebra. That would be quite a mess I think. Oleg Alexandrov 5 July 2005 15:56 (UTC)

title: Talk:Advanced modular arithmetic theory


text:
Aperture (disambiguation)

The aperture of an optical system is the opening that limits the amount of light that can pass through.

Aperture may also refer to:

Science and technology

 * In anatomy, a number of apertures in the human body:
 * Apertura nasalis posterior
 * Apertura pelvis inferior
 * Apertura pelvis minoris
 * Apertura pelvis superior
 * Apertura thoracis inferior
 * Apertura thoracis superior
 * Lateral aperture (foramen of Luschka), an opening in each lateral extremity of the lateral recess of the fourth ventricle of the human brain
 * Median aperture (foramen of Magendie), which drains cerebrospinal fluid (CSF) from the fourth ventricle into the cisterna magna
 * Antenna aperture, a physical parameter of an antenna
 * Aperture (mollusc), the main opening in the shell of a gastropod or scaphopod (mollusc)
 * Aperture (botany), a weaker spot in the wall of a pollen grain
 * Numerical aperture is a parameter used to describe optical systems
 * Aperture (computer memory), a region of the physical address space that opens access to a particular device or memory unit
 * Aperture (software), an image organization and editing program for photographers, sold by Apple Inc.

Other uses

 * Aperture (magazine), a long-running art photography magazine
 * "Aperture" (2020), a song by Emily Blue
 * Aperture (typography), an enclosed loop in a typographical symbol, such as the lower part of the letter 'a'
 * Aperture Foundation, a photography nonprofit
 * Aperture Science, a fictional organization featured in the Portal (series) of video games
 * Aperture Peak

title: Aperture (disambiguation)


text:
Duddon Estuary

The Duddon Estuary is the sandy, gritty estuary of the River Duddon that lies between Morecambe Bay and the North Lonsdale coast.

The River Duddon and its estuary form part of the boundary of the historic county of Lancashire.

It opens into the Irish Sea to the north of the Furness peninsula; Walney Island forming part of its southern edge. Its 28 miles (45 km) of shoreline enclose an area of 13 square miles (35 km2), making it the second largest estuary in Cumbria after the Solway Firth and one of the six main estuaries in the historic county of Lancashire.

The main settlements alongside the Duddon estuary are Haverigg, Millom, Foxfield, Kirkby-in-Furness, Askam and Ireleth and Barrow-in-Furness.

Biological importance
The estuary as a whole was designated as a Site of Special Scientific Interest (SSSI) in 1990 with the amalgamation of five previously separate SSSIs: Duddon Sands, Sandscale Haws, North Walney, Hodbarrow Lagoon and Haverigg Haws. It is a Ramsar site.

Fauna
The Duddon Estuary is significant for natterjack toads. It supports one fifth of the national population of the rare amphibian that is only found at 50 sites in the UK, of which five are in the Duddon Estuary.

Birds
The Duddon Estuary is an Important Bird Area. Species to be seen include pintail, red knot and common redshank with wintering waterfowl including common shelduck, red-breasted mergansers, Eurasian oystercatchers, ringed plover, dunlin and Eurasian curlew.

In 1998 it was designated a Special Protection Area (SPA) under the Birds Directive. It qualified under three criteria:
 * regularly there are over 20,000 wintering waterfowl.
 * breeding population of sandwich terns
 * overwintering populations of knot, pintail and redshank; populations on passage of ringed plover and sanderling.

In 2015, before the United Kingdom European Union membership referendum, there were consultations on a successful proposal that Morecambe Bay and Duddon Estuary be combined in a new SPA. This would amalgamate the existing Morecambe Bay and Duddon Estuary SPAs and would add marine areas identified as being used by foraging terns.

Flora
The estuary is botanically rich with salt marsh, sand dune and shingle communities, including a nationally rare shingle vegetation community at Haverigg Haws and North Walney.

Shingle species include sea sandwort, spear-leaved orache, sea rocket and sea kale. All the dune grasslands at Sandscale Haws, Haverigg Haws and North Walney support a rich flora with the rare dune helleborine.

Threats
Development has had no significant effect on the nature conservation interest of the estuary, but it is at risk from coastal defence works, grazing by agricultural stock, sea level rise, recreational pressure and bait digging.

title: Duddon Estuary


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Portal:Biology/Did you know

...Archive

title: Portal:Biology/Did you know


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Category:Optical fiber

This category is for the various types of optical fiber.

 * For information about the science and technology of optical fiber, see Category:Fiber optics.

title: Category:Optical fiber


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Template:Portals

What are Portal? · Portal:Contents/Portals

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Vasco da Gama (crater)

Vasco da Gama is a lunar impact crater that is located near the western limb of the Moon. It is named for the Portuguese explorer Vasco da Gama. It lies to the south of the walled plain Einstein, due south of Dalton. Attached to the southwestern rim of Vasco da Gama is the somewhat smaller crater Bohr.

The northern and southern parts of the rim of Vasco da Gama have been heavily eroded by impacts, with the western and eastern rims being more intact. The smaller crater Vasco da Gama S intrudes into the southeastern rim, and just to its southwest is Vasco da Gama T. Near the northeastern rim is Vasco da Gama B. There is a low central massif near the midpoint of the interior.

The interior floor of satellite crater Vasco da Gama R contains a system of rilles designated the Rimae Vasco da Gama.

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 Vasco da Gama.

title: Vasco da Gama (crater)


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Neoarchean

The Neoarchean (also spelled Neoarchaean) is the last geologic era in the Archean Eon that spans from 2800 to 2500 million years ago—the period being defined chronometrically and not referencing a specific level in a rock section on Earth. The era is marked by major developments in complex life and continental formation.

Complex life
This era saw the rise of oxygen in the atmosphere after oxygenic photosynthesis evolved in cyanobacteria as early as the Mesoarchean era. The environmental changes that occurred in the Neoarchean such as its developing atmospheric and soil compositions drastically differentiated the era from others in its encouragement of microbial metabolisms to evolve and diversify. The era could have also seen pre-biotic organic molecules being brought to Earth through meteorites, comets, or through abiotic reactions. The growth of juvenile continental crust as well as the onset of plate tectonics in the Archean allowed for the colonization of a larger variety of niches by microorganisms through increasing the number of rock types present and thereby increasing the surface's chemical diversity. Some noted metabolisms were able to flourish due to changes in the availability of certain metals while others faced famine: an increase in copper present in the environment in the Neoarchean likely favored aerobic metabolisms.

Oxygenic photosynthesis may have been limited earlier in the Archean era from a lack of phosphorus stemming from poor biological recycling in anaerobic conditions. This issue was alleviated in the Neoarchean with the abundance of phosphorus in magmatic rocks, which when combined with other evolving geodynamics such as increasing organic matter burial and higher oxidative states in volcanic sulfur and magmatic iron contributed to a large buildup of oxygen in the atmosphere, leading to the Great Oxidation Event in the Paleoproterozoic era.

The earliest evidence of microbial oxidation of sulphur, dating to approximately 2.52 Ga, comes from the Gamohaan Formation of South Africa, revealing that sulphur-oxidising bacteria had evolved prior to the Great Oxidation Event.

The earliest potential eukaryote fossils come from Neoarchaean deposits in South Africa dating to 2.8 to 2.7 Ga, resembling present day siphonalean microalgae. However, the identity of these microfossils as eukaryotes is highly controversial and remains disputed.

Continental formation
During this era, the supercontinent Kenorland is proposed to have formed about 2.7 billion years ago. Kenorland is of particular interest due to it containing deposits of volcanic-hosted massive sulphide, gold, and uranium found in the Canadian Shield. With new research, the validity of Kenorland has been questioned in favor of other Neoarchean supercontinent proposals Superia or Vaalbara. Improved geologic knowledge suggests that a part of Kenorland, specifically the Churchill Province, was instead a continental development that formed after the Neoarchean era, Nuna, closer to 1.9 billion years ago. This challenge to the reconstruction is based on research studying northern Kenorland's Paleoproterozoic cover as well as the suture between the Rae and Hearne cratons.

The supercontinent cycle can be studied through patterns that describe how Earth's crust and its mineral deposits were preserved over time since Pangaea. Plate tectonics, having developed earlier in the Archean eon, produced the force necessary for metamorphism and magmatic activity which greatly contributed to these continental changes. Research on how the supercontinents broke apart and combined into different configurations is involved in linking together deep-interior and surface-level processes as well as the assessment of contrasting models of early Paleoproterozoic geodynamic activity.

title: Neoarchean