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Fish protein powder	Categories	The two basic categories used to classify fish protein powders are dependent on the levels of protein, fat, mineral and carbohydrate contained in the powder. The minerals are mostly naturally occurring, organic complexes of magnesium, calcium and phosphorus. The spray drying process may utilize other minerals and carbohydrates to improve flow characteristics of the final product thus altering the natural balance. Powders will all have a residual moisture content in the 4-8% range.
Fish protein powder	Categories	Fish protein concentrate (FPC) - is a powder concentrate with medium level of protein (50-70%) and will contain some level of fat/oil (1-20%) in the powder form as well. Fish protein isolate (FPi) - where the product contains less than 1% fat/oil and more than 90% protein. New manufacturing techniques are also producing hybrid FPi products where the fat/oil content is very low, (<0.3%) with the protein levels in the 80% range. The hybrid FPi does not reach 90% protein (often a definition point for an isolate) as the natural minerals are not removed and thus represent up to 15% of the final mass balance.
Fish protein powder	Peptides vs proteins and amino acids in the digestive tract	Any animal that consumes a whole protein must break down and digest the protein order to absorb the nutrients. For humans this begins with chewing and the addition of saliva enzymes, followed by acid and protease enzyme digestion in the stomach, whereby the end result is a peptide or amino acid fraction ready for uptake into the blood stream via the small intestine. Research has confirmed that most animals have more Peptide receptors in the gut and lower intestine than they do free amino acid receptors - as such the peptide form of fish protein powder is most conducive for optimal nutritional benefits.Hygienic production of fish protein powder mimics these natural digestion steps, and pending the degree of hydrolysis, the protein powder will actually be a partial or complete peptide powder, ready for immediate absorption in the intestine.
Fish protein powder	Nutritional aspects	Significant elements of the nutritional science of fish protein powders centers around the bioactive and antioxidant properties of the peptide fractions produced during hydrolysis and their ability to have a positive impact on many conditions including gastrointestinal issues associated with irritable bowel syndrome (IBS) and Crohn's disease - as well reduction effects on hypertension and fast absorption functionality promotes the addition of lean muscle mass to humans consuming the products. Further studies showed that peptides in fish protein powders can minimize injurious effects of anti-inflammatory pain drugs. The University of Maryland School of Medicine concluded that certain peptide fractions from fish may inhibit prostate cancer and possibly other cancers from spreading.Additional benefits of fish protein powders are centered around diet needs of various subsets of the human population. Individuals who have lactose intolerance, milk allergy, gluten intolerance or coeliac disease (aka Celiac's) require alternate protein sources.The hydrolyzed nature of fish protein powder (low molecular weight profile) leads it to be used in hypoallergenic applications such as infant formulas. There is no evidence that infants who have a high risk of having an allergy to cows milk should be fed hydrolyzed infant formula instead of breast milk for allergy prevention. For infants who have a high-risk of a cows milk allergy but cannot be fed breast milk, there is low-quality evidence suggesting that hydrolyzed protein-based formula may reduce the risk of a cows milk allergy compared to cow milk protein formula.
Caffeine-Free Pepsi	Caffeine-Free Pepsi	Caffeine-Free Pepsi is a version of the cola Pepsi that omits the caffeine that is customarily part of a cola. It was introduced under the brand name "Pepsi Free" in 1982 by PepsiCo. It was 99.7 percent caffeine free. A sugar-free variant was also introduced and known as "Diet Pepsi Free," The "Pepsi Free" name itself was phased out in 1987, and today these colas are known simply as "Caffeine-Free Pepsi" and "Caffeine-Free Diet Pepsi."
Caffeine-Free Pepsi	Background	When it was first introduced, Caffeine-Free Pepsi's label background was red, but to avoid any confusion with Coca-Cola, the background color was changed to gold in 1987. As part of Pepsi's changing their background to blue in 1998, Pepsi Caffeine Free's background changed to blue with the letters outlined in gold. In 2009, the caffeine-free version reverted to a gold background. Caffeine-Free Coca-Cola labels also have a gold background. The logo letters are bordered in red for the regular variety; in the case of the diet variety, they are entirely red in color.
Caffeine-Free Pepsi	Background	When introduced, Pepsi Free was available in cans, 2-liter bottles, and 20-ounce glass bottles. Caffeine-Free Pepsi is currently available in cans, 16 oz. plastic bottles and 2 liters, though availability varies from store to store (for instance, 16 oz. bottles are typically only available in convenience stores, and some grocers may only have the product in 12 oz cans, if they carry it at all).
Caffeine-Free Pepsi	In popular culture	Two cans of Pepsi Free are seen, at separate times, in the 1983 film Mr. Mom. Two-liter bottles and six-packs of cans of Pepsi Free appear in a refrigerator case behind Sylvester Stallone's character in the grocery store scene in the 1986 film Cobra.
Caffeine-Free Pepsi	In popular culture	Pepsi Free was the subject of a scene in the 1985 film Back to the Future. Upon entering a café in 1955, Marty McFly (Michael J. Fox) asks for a Tab (Coca-Cola's first version of a sugar-free soft drink, which was not available until 1963) and is told that he cannot have a "tab," unless he orders something. He then asks for a Pepsi Free (also not available in the 1950s) and is told, "If you want a Pepsi, pal, you're gonna pay for it!" ("Free" is here being mistaken for gratis.) Finally, he asks for "something without any sugar in it," and is served black coffee.
Caffeine-Free Pepsi	In popular culture	A can of Diet Pepsi Free can be seen beside Marty's alarm clock towards the beginning of the movie when Doc (Christopher Lloyd) calls him to remind him to meet him at the mall. The can is also seen toward the end of the movie when Marty wakes up in the morning at his house in 1985.
Pronunciation of GIF	Pronunciation of GIF	The pronunciation of GIF, an acronym for the Graphics Interchange Format, has been disputed since the 1990s. Popularly rendered in English as a one-syllable word, the acronym is most commonly pronounced (listen) (with a hard g as in gift) or (listen) (with a soft g as in gem), differing in the phoneme represented by the letter G. Many public figures and institutions have taken sides in the debate; Steve Wilhite, the creator of the image file format, gave a speech at the 2013 Webby Awards arguing for the soft-g pronunciation. Others have pointed to the term's origin from abbreviation of the hard-g word graphics to argue for the other pronunciation.
Pronunciation of GIF	Pronunciation of GIF	The controversy stems partly from the fact that there is no general rule for how the letter sequence gi is to be pronounced; the hard g prevails in words such as gift, while the soft g is used in others such as ginger. In addition, some speakers enunciate each letter in the acronym, producing (listen). English dictionaries generally accept both main alternatives as valid, and linguistic analyses show no clear advantage for either based on the pronunciation frequencies of similar English words. The pronunciation of the acronym can also vary in languages other than English.
Pronunciation of GIF	Background	The Graphics Interchange Format (GIF) is an image file format developed in 1987 by Steve Wilhite at the American online service provider CompuServe. GIFs are popularly used to display short, looped animations. The acronym GIF, commonly pronounced as a monosyllable, has a disputed pronunciation. Some individuals pronounce the word with a hard g, as in (listen), whereas others pronounce it with a soft g, as in (listen). A minority prefer to enunciate each letter of the acronym individually, creating the pronunciation (listen).Wilhite and the team who developed the file format included in the technical specifications that the acronym was to be pronounced with a soft g. In the specifications, the team wrote that "choosy programmers choose ... 'jif'", in homage to the peanut butter company Jif's advertising slogan of "choosy moms choose Jif". According to ABC News, the debate stretches as far back as 1994, with an author of an encyclopedia of image formats stating that "most people" seem to prefer the hard g pronunciation over his preferred soft g.
Pronunciation of GIF	Background	Other languages In French, the acronym tends to be pronounced [ʒif] (listen), with the voiced postalveolar fricative, [ʒ], as in the j in the French joie or the s in the English measure or vision, even though [dʒ], which does not occur in native vocabulary, tends to be retained in English loanwords (such as jeans). Some languages lack English's soft and hard g sounds in their phonologies; Spanish and Finnish, for example, lack [ʒ] in their native words. In Norwegian, GIF is pronounced with a hard g, [ɡ], unlike native words, for which the sequence ⟨gi⟩ would be pronounced with a voiced palatal approximant, [j], like the y in English yes.
Pronunciation of GIF	Analysis of the dispute	Cause In English, the linguistic controversy stems partly from the fact that there is no general rule for how the letter sequence gi is to be pronounced; the hard g prevails in words such as gift, while the soft g is used in others, such as ginger. In Old English, g would make the soft g sound as well as y's consonant sound, and when the hard g was added, both its hard and soft variations persisted when followed by i.An analysis of 269 words by linguist Michael Dow found near-tied results on whether a hard or soft g was more appropriate based on other English words; the results varied somewhat depending on what parameters were used. Of the 105 words that contained gi somewhere in the word, 68 used the soft g while only 37 employed its counterpart. However, the hard g words were found to be significantly more common in everyday English; comparatively obscure words like flibbertigibbet and tergiversate, both pronounced with a soft g, were included in the list of 68 soft gi words. When the prevalence of each word was taken into account, it was found that the hard and soft g appeared in nearly equal frequencies in gi words. No clear favorite was found by only using the words that begin with gi, nor by only using words with one syllable such as gift and gin.
Pronunciation of GIF	Analysis of the dispute	In her coverage of Dow's piece, Canadian linguist Gretchen McCulloch theorizes that since the hard and soft g in this context are used with near-equal frequency, when a person first encounters the word GIF, they make a guess akin to flipping a coin by comparing it to other words they have encountered in the past. Once they have a favorite one way or the other, the notion is solidified—leading McCulloch to comment that this "probably means we'll be fighting the gif pronunciation war for generations to come".
Pronunciation of GIF	Analysis of the dispute	Arguments A 2019 analysis by linguist Marten van der Meulen found that the most common arguments employed online over the pronunciation of GIF are "system" arguments, which support one side of the debate by contending that the pronunciation should flow from a consistent rule of language. One example of this would be the "system acronym" argument: the idea that because the letter G in GIF stands for the word graphics, it ought to be pronounced in the acronym with the same phoneme as in the word, i.e. with a hard g. This particular argument is sometimes accompanied by the quip that if the acronym were to be pronounced with a soft g, the word should be pronounced likewise, as ("jraphics"). A rebuttal to this argument is that acronyms are not required to follow the pronunciations of their root words. For example, the letter u in the word scuba (listen)—an acronym for self-contained underwater breathing apparatus—is pronounced even though its deriving word, underwater, is pronounced instead with . A similar acronym discrepancy arises with NASA (National Aeronautics and Space Administration, pronounced (listen)).Another example of a "system" argument is frequency analysis, which examines how many other English words employ hard or soft g pronunciations in other situations, similar to Dow's analysis. After Steve Wilhite announced his opinion that the soft g pronunciation was the only correct form, there was significant chatter on social media and in the press on both sides of the issue. An article by Casey Chan, writing for Gizmodo, argued that Wilhite was wrong because soft g words followed by if should be spelled with the letter j, such as the "jiffy" in "Jiffy Lube" and "be back in a jiffy", as well as the peanut butter company Jif.The next most common argument found in van der Meulen's analysis was an argument that cited an authority, usually Wilhite, as the creator of the file format. After Wilhite announced his support for the soft g pronunciation, many recognized him as the authority on the pronunciation of the word due to his creation of its format. Wilhite is the most commonly cited authority for the pronunciation of GIF; 65.2 percent of surveyed arguments citing an authority favored a soft g. Some, including Casey Chan, cited U.S. President Barack Obama in supporting the hard g; others cited various dictionaries, or software assistants such as Siri as authorities for GIF's pronunciation.
Pronunciation of GIF	Analysis of the dispute	Polling A 2014 Mashable poll of more than 30,000 people worldwide found that seven in ten used the hard g. Van der Meulen's analysis found that 57.2 percent of users who offered an opinion supported the hard g, while 31.8 percent favored the soft g. The analysis also found that 8.2 percent of users support both pronunciations, while favoring the soft g, and 2.8 percent favored enunciating each letter.An informal poll of developers on Stack Overflow showed that 65.6 percent of respondents favored the hard g pronunciation, while 26.3 percent used the soft g, 6 percent sounded out every letter, and 2 percent employed a different pronunciation altogether. However, an analysis from The Economist argued that the disparities in the results were exaggerated by sampling bias; the article commented that while the countries where the hard g is used make up 45 percent of the world's population, respondents from those countries comprised 79 percent of the sample. When the populations of each country were adjusted for, the analysis found that hard g still led, albeit by a narrower margin of 44 percent to 32 percent for soft g. In addition, this adjustment brought the popularity of pronouncing each letter up to 21 percent; this variation is common in Asian countries, where it is employed by half of Chinese respondents and 70 percent of South Korean respondents. Developed countries as a whole tended to favor the hard g pronunciation.
Pronunciation of GIF	Analysis of the dispute	Dictionaries Dictionary.com lists both the hard and soft g pronunciations for GIF, indicating the latter as the primary pronunciation, while the Cambridge Dictionary of American English and the Cambridge Advanced Learner's Dictionary offer only the hard g pronunciation. The online Merriam-Webster and Lexico dictionaries list both pronunciations. The 2005 edition of the New Oxford American Dictionary gave only the soft g pronunciation; the 2010 edition of the Oxford Dictionary of English listed both pronunciations, listing the soft g first. The French Petit Robert and Petit Larousse list only [ʒif] in their entries. In the Norwegian Academy's dictionary of the Norwegian language, the pronunciation is transcribed with a hard g, as [gifː].
Pronunciation of GIF	Incidents	In May 2013, Wilhite was presented with a lifetime achievement award at the annual Webby Awards honoring excellence on the Internet. Upon accepting the award at the ceremony, Wilhite displayed a five-word slide that simply read, in all caps: "It's pronounced 'jif' not 'gif'". Here, jif refers to the soft g pronunciation. Following the speech, Wilhite told The New York Times: "The Oxford English Dictionary accepts both pronunciations. They are wrong. It is a soft g ... End of story."The audience attending the ceremony reacted positively to the short speech, but it generated controversy online, with some commentators pushing back against Wilhite's pronunciation. Van der Meulen remarked that this "seems to be the first ever coiner of a word (or acronym, to be more specific) who gave usage advice about his own creation". More than 17,000 tweets were made in the aftermath of the speech, making "GIF" a trending topic, and more than 50 news articles were written on the incident. The Columbia Journalism Review remarked three years later that the debate seemed to peak with this incident. The peanut butter company Jif responded to a tweet asking how they were feeling following the speech, commenting, "We're nuts about him today." Seven years later, Jif performed a publicity stunt with GIF-hosting platform Giphy. The two companies released a joint statement, arguing that the correct pronunciation employs a hard g and releasing limited-time jars of peanut butter labeled "GIF" instead of "JIF".In October 2013, The New York Times faced some light criticism on social media for an article that began with the words, "A GIF, pronounced jif, is a compressed image file format invented in 1987." The article included a link to an earlier article from the newspaper, covering Wilhite's speech and the quote he gave them. In December 2013, Alex Trebek, the host of game show Jeopardy!, attracted media attention when the final clue of the episode referenced Wilhite's presentation and opinion on the pronunciation. Trebek read out the responses of contestants using a soft g when the word "GIF" appeared in the correct responses of all three contestants. In the past, Trebek had pronounced each letter individually, to remain neutral.In June 2014, Barack Obama, then President of the United States, opined that the acronym should be pronounced with a hard g when prompted in a conversation with David Karp, the founder of Tumblr. Miles Klee of The Daily Dot highlighted an April 2013 post on the White House's Tumblr blog, which included a humorous infographic with the text "animated GIFs (hard 'g')".
Mazovia encoding	Mazovia encoding	Mazovia encoding is a character set used under DOS to represent Polish text. The character set derives from code page 437, with specific positions modified to accommodate Polish letters. Notably, the Mazovia encoding maintains the block graphic characters from code page 437, distinguishing it from IBM's later official Central European code page 852, which failed to preserve all block graphics, leading to incorrect display in programs such as Norton Commander.
Mazovia encoding	Mazovia encoding	The Mazovia encoding was designed in 1984 by Jan Klimowicz of IMM. It was designed as part of a project to develop and produce a Polish IBM PC clone codenamed "Mazovia 1016". The code page was specifically optimized for the peripheral devices commonly used with the Mazovia 1016 computer, including a graphics card with dual switchable graphics, a keyboard with US English and Russian layouts, and printers with Polish fonts. The Mazovia encoding gained widespread acceptance and distribution in Poland when the Polish National Bank (NBP) adopted it as a standard in 1986. The NBP played a significant role in facilitating the production of compatible computers by Ipaco, which utilized Taiwanese components under the guidance of Zbigniew Jakubas and Krzysztof Sochacki.
Mazovia encoding	Mazovia encoding	Some ambiguity exists in the official code page assignment for the Mazovia encoding: PTS-DOS and S/DOS support this encoding under code page 667 (CP667). The same encoding was also called code page 991 (CP991) in some Polish software, however, the FreeDOS implementation of code page 991 seems not to be identical to this original encoding.
Mazovia encoding	Mazovia encoding	The DOS code page switching file NECPINW.CPI for NEC Pinwriters supports the Mazovia encoding under both code pages 667 and 991. FreeDOS has meanwhile introduced support for the original Mazovia encoding under code page 790 (CP790) as well. The Fujitsu DL6400 (Pro) / DL6600 (Pro) printers support the Mazovia encoding as well. This encoding is known as code page 3843 in Star printers.
Mazovia encoding	Character set	Each character is shown with its equivalent Unicode code point. Only the second half of the table (128–255) is shown, all of the first half (0–127) being the same as ASCII and code page 437.
Mazovia encoding	Character set	Several variants of this encoding exists: Mazovia 157 (ś is at 9D instead of 9E) Fido Mazovia (ć is at 0x87 instead of 8D and Ć is at 0x80 instead of 0x95) FreeDOS Mazovia (złoty sign at 9B). FreeDOS supports this variant under code page 991, although the original definition of code page 991, which pre-dates FreeDOS, appears to have been identical to code page 667 / 790.These variants are not fully compliant with the definition of code page 667 / 790 and should therefore not be associated with these numbers.
Radical 89	Radical 89	Radical 89 or radical double x (爻部) meaning "trigrams" is one of the 34 Kangxi radicals (214 radicals in total) composed of 4 strokes. In the Kangxi Dictionary, there are 16 characters (out of 49,030) to be found under this radical. This radical does not exist in the Table of Indexing Chinese Character Components predominantly adopted by Simplified Chinese dictionaries published in mainland China.
Radical 89	Literature	Fazzioli, Edoardo (1987). Chinese calligraphy : from pictograph to ideogram : the history of 214 essential Chinese/Japanese characters. calligraphy by Rebecca Hon Ko. New York: Abbeville Press. ISBN 0-89659-774-1. Lunde, Ken (Jan 5, 2009). "Appendix J: Japanese Character Sets" (PDF). CJKV Information Processing: Chinese, Japanese, Korean & Vietnamese Computing (Second ed.). Sebastopol, Calif.: O'Reilly Media. ISBN 978-0-596-51447-1.
Allotropes of iron	Allotropes of iron	At atmospheric pressure, three allotropic forms of iron exist, depending on temperature: alpha iron (α-Fe, ferrite), gamma iron (γ-Fe, austenite), and delta iron (δ-Fe). At very high pressure, a fourth form exists, epsilon iron (ε-Fe, hexaferrum). Some controversial experimental evidence suggests the existence of a fifth high-pressure form that is stable at very high pressures and temperatures.The phases of iron at atmospheric pressure are important because of the differences in solubility of carbon, forming different types of steel. The high-pressure phases of iron are important as models for the solid parts of planetary cores. The inner core of the Earth is generally assumed to consist essentially of a crystalline iron-nickel alloy with ε structure. The outer core surrounding the solid inner core is believed to be composed of liquid iron mixed with nickel and trace amounts of lighter elements.
Allotropes of iron	Standard pressure allotropes	Alpha iron (α-Fe) Below 912 °C (1,674 °F), iron has a body-centered cubic (bcc) crystal structure and is known as α-iron or ferrite. It is thermodynamically stable and a fairly soft metal. α-Fe can be subjected to pressures up to ca. 15 GPa before transforming into a high-pressure form termed ε-Fe discussed below.
Allotropes of iron	Standard pressure allotropes	Magnetically, α-iron is paramagnetic at high temperatures. However, below its Curie temperature (TC or A2) of 771 °C (1044K or 1420 °F), it becomes ferromagnetic. In the past, the paramagnetic form of α-iron was known as beta iron (β-Fe). Even though the slight tetragonal distortion in the ferromagnetic state does constitute a true phase transition, the continuous nature of this transition results in only minor importance in steel heat treating. The A2 line forms the boundary between the beta iron and alpha fields in the phase diagram in Figure 1.
Allotropes of iron	Standard pressure allotropes	Similarly, the A2 boundary is of only minor importance compared to the A1 (eutectoid), A3 and Acm critical temperatures. The Acm, where austenite is in equilibrium with cementite + γ-Fe, is beyond the right edge in Fig. 1. The α + γ phase field is, technically, the β + γ field above the A2. The beta designation maintains continuity of the Greek-letter progression of phases in iron and steel: α-Fe, β-Fe, austenite (γ-Fe), high-temperature δ-Fe, and high-pressure hexaferrum (ε-Fe).
Allotropes of iron	Standard pressure allotropes	The primary phase of low-carbon or mild steel and most cast irons at room temperature is ferromagnetic α-Fe. It has a hardness of approximately 80 Brinell. The maximum solubility of carbon is about 0.02 wt% at 727 °C (1,341 °F) and 0.001% at 0 °C (32 °F). When it dissolves in iron, carbon atoms occupy interstitial "holes". Being about twice the diameter of the tetrahedral hole, the carbon introduces a strong local strain field.
Allotropes of iron	Standard pressure allotropes	Mild steel (carbon steel with up to about 0.2 wt% C) consists mostly of α-Fe and increasing amounts of cementite (Fe3C, an iron carbide). The mixture adopts a lamellar structure called pearlite. Since bainite and pearlite each contain α-Fe as a component, any iron-carbon alloy will contain some amount of α-Fe if it is allowed to reach equilibrium at room temperature. The amount of α-Fe depends on the cooling process.
Allotropes of iron	Standard pressure allotropes	A2 critical temperature and induction heating β-Fe and the A2 critical temperature are important in induction heating of steel, such as for surface-hardening heat treatments. Steel is typically austenitized at 900–1000 °C before it is quenched and tempered. The high-frequency alternating magnetic field of induction heating heats the steel by two mechanisms below the Curie temperature: resistance or Joule heating and ferromagnetic hysteresis losses. Above the A2 boundary, the hysteresis mechanism disappears and the required amount of energy per degree of temperature increase is thus substantially larger than below A2. Load-matching circuits may be needed to vary the impedance in the induction power source to compensate for the change.
Allotropes of iron	Standard pressure allotropes	Gamma iron (γ-Fe) When heating iron above 912 °C (1,674 °F), its crystal structure changes to a face-centered cubic (fcc) crystalline structure. In this form it is called gamma iron (γ-Fe) or austenite. γ-iron can dissolve considerably more carbon (as much as 2.04% by mass at 1,146 °C). This γ form of carbon saturation is exhibited in austenitic stainless steel.
Allotropes of iron	Standard pressure allotropes	Delta iron (δ-Fe) Peculiarly, above 1,394 °C (2,541 °F) iron changes back into the bcc structure, known as δ-Fe. δ-iron can dissolve as much as 0.08% of carbon by mass at 1,475 °C. It is stable up to its melting point of 1,538 °C (2,800 °F). δ-Fe cannot exist above 5.2 GPa, with austenite instead transitioning directly to a molten phase at these high pressures.
Allotropes of iron	High pressure allotropes	Epsilon iron / Hexaferrum (ε-Fe) At pressures above approximately 10-13 GPa and temperatures up to around 700 K, α-iron changes into a hexagonal close-packed (hcp) structure, which is also known as ε-iron or hexaferrum; the higher-temperature γ-phase also changes into ε-iron, but generally requires far higher pressures as temperature increases. The triple point of hexaferrum, ferrite, and austenite is 10.5 GPa at 750 K. Antiferromagnetism in alloys of epsilon-Fe with Mn, Os and Ru has been observed.
Allotropes of iron	High pressure allotropes	Experimental high temperature and pressure An alternate stable form, if it exists, may appear at pressures of at least 50 GPa and temperatures of at least 1,500 K; it has been thought to have an orthorhombic or a double hcp structure. As of December 2011, recent and ongoing experiments are being conducted on high-pressure and superdense carbon allotropes.
Allotropes of iron	Phase transitions	Melting and boiling points The melting point of iron is experimentally well defined for pressures less than 50 GPa.
Allotropes of iron	Phase transitions	For greater pressures, published data (as of 2007) put the γ-ε-liquid triple point at pressures that differ by tens of gigapascals and 1000 K in the melting point. Generally speaking, molecular dynamics computer simulations of iron melting and shock wave experiments suggest higher melting points and a much steeper slope of the melting curve than static experiments carried out in diamond anvil cells.The melting and boiling points of iron, along with its enthalpy of atomization, are lower than those of the earlier group 3d elements from scandium to chromium, showing the lessened contribution of the 3d electrons to metallic bonding as they are attracted more and more into the inert core by the nucleus; however, they are higher than the values for the previous element manganese because that element has a half-filled 3d subshell and consequently its d-electrons are not easily delocalized. This same trend appears for ruthenium but not osmium.
Allotropes of iron	Phase transitions	Structural phase transitions The exact temperatures at which iron will transition from one crystal structure to another depends on how much and what type of other elements are dissolved in the iron. The phase boundary between the different solid phases is drawn on a binary phase diagram, usually plotted as temperature versus percent iron. Adding some elements, such as Chromium, narrows the temperature range for the gamma phase, while others increase the temperature range of the gamma phase. In elements that reduce the gamma phase range, the alpha-gamma phase boundary connects with the gamma-delta phase boundary, forming what is usually called the Gamma loop. Adding Gamma loop additives keeps the iron in a body-centered cubic structure and prevents the steel from suffering phase transition to other solid states.
Charlotte (cake)	Charlotte (cake)	A charlotte is a type of bread pudding that can be served hot or cold. It is also referred to as an "icebox cake". Bread, sponge cake, crumbs or biscuits/cookies are used to line a mold, which is then filled with a fruit puree or custard. The baked pudding could then be sprinkled with powdered sugar and glazed with a salamander, a red-hot iron plate attached to a long handle, though modern recipes would likely use more practical tools to achieve a similar effect.
Charlotte (cake)	Charlotte (cake)	The variant charlotte russe also called charlotte parisienne, created by the French chef Antonin Carême, uses a mold lined with ladyfingers and filled with Bavarian cream. Classically, stale bread dipped in butter was used as the lining, but sponge cake or ladyfingers may be used today. The filling may be covered with a thin layer of similarly flavoured gelatin.
Charlotte (cake)	History	The charlotte is known to have existed by the late-18th century. In 1796, The New-York Magazine published a poem by Joel Barlow called The Hasty-Pudding which included the following lines: Some have claimed that it was a tribute to Britain's Queen Charlotte.In 1815, Marie-Antoine Carême claims to have thought of charlotte à la parisienne "pendant mon établissement", presumably in 1803, when he opened his own pastry shop.: 446 The earliest known English recipe is from the 1808 London edition of Maria Rundell's New System of Domestic Cookery: Cut as many very thin slices of white bread as will cover the bottom and line the sides of a baking dish, but first rub it thick with butter. Put apples, in thin slices, into the dish, in layers, till full, strewing sugar between, and bits of butter. In the mean time, soak as many thin slices of bread as will cover the whole, in warm milk, over which lay a plate, and a weight to keep the bread close on the apples. Bake slowly three hours. To a middling sized dish use half a pound of butter in the whole.
Charlotte (cake)	History	In Carême's 1815 Le Pâtissier royal parisien, he mentions many varieties of charlotte: à la parisienne, à la française, à l'italienne, aux macarons d'avelines, aux gaufres aux pistaches, de pommes, de pomme d'api, d'abricots, de pêches, de pommes glacée aux abricots, de pommes au beurre, parisienne à la vanille, de pommes; he mentions à la russe as the name used by others for what he called à la parisienne.
Charlotte (cake)	Types	There are many variants. Most charlottes are served cool, so they are more common in warmer seasons. Fruit charlottes usually combine a fruit purée or preserve, like raspberry or pear, with a custard filling or whipped cream. Charlottes are not always made with fruit; some, notably charlotte russe, use custard or Bavarian cream, and a chocolate charlotte is made with layers of chocolate mousse filling. The Algerian charlotte is made with honey, dates, orange rind, and almonds.The 19th-century Russian sharlotka is a baked pudding with layers of brown bread and apple sauce, and has since evolved into a simple dessert of chopped apples baked in a sweet batter.
Charlotte (cake)	Types	Charlotte russe Charlotte russe or charlotte à la russe is a cold dessert of Bavarian cream set in a mold lined with ladyfingers.A simplified version of charlotte russe was a popular dessert or on-the-go treat sold in candy stores and luncheonettes in New York City, during the 1930s, 1940s, and 1950s. It consisted of a paper cup filled with yellow cake and whipped cream topped with half a maraschino cherry. The bottom of the cup is pushed up to eat.Charlotte royale is made with the same filling as a Charlotte russe, but the ladyfingers are replaced by slices of Swiss roll.
Charlotte (cake)	Etymology	The earliest attestation of "charlotte" is in a New York magazine in 1796. Its origins are unclear. It may come from the woman's name. One etymology suggests it is a corruption of the Old English word charlyt, a kind of custard, or charlets, a meat dish.It is often claimed that Carême named it charlotte after one of the various foreign royals he served, but the name appears years earlier.
Charlotte (cake)	Etymology	Carême's preferred name for charlotte à la russe was charlotte à la parisienne, and he says (in 1815) that "others" prefer to call it russe,: 446 so it is unlikely that he named it russe for Czar Alexander I as has been proposed.
Nailset	Nailset	A nailset or nail punch is a hand tool used for driving the exposed head of a nail or pin below the surface of a piece of wood, such as when installing decorative moulding or face-fastening wood flooring.
Nailset	Nailset	Though they vary in design, nailsets are typically made from a hard round or square steel rod which tapers at one end to a flat or slightly hollowed tip. The tip is placed against the head of the nail, while the other end of the nailset is struck with a hammer. Nailsets come with different sized tips suited to different sized nail heads.
't	't	In the Dutch language, the word 't (Dutch pronunciation: [ət]) is a contraction of the article "het", meaning "the". 't can be found as a tussenvoegsel, a word that is positioned between a person's first and last name. Careful writers should use an apostrophe (U+2019 ’ ) in front of the t – and not confuse it with a left quotation mark (U+2018 ‘ ) and put a space before the apostrophe.
't	Examples	Dirk van 't Klooster Evert-Jan 't Hoen Gerard 't Hooft Haas Visser 't Hooft in 't Veld (surname) Bart Spring in 't Veld Sophie in 't Veld Jacobus Henricus van 't Hoff John van 't Schip Maarten 't Hart Tom van 't Hek Van 't Hof (surname) Van 't Wout (surname) Willem Visser 't Hooft Youp van 't Hek
Kirsanov reaction	Kirsanov reaction	The Kirsanov reaction is a method for the synthesis of certain organophosphorus compounds. In this reaction a tertiary phosphine is combined with a halogen and then an amine to give the iminophosphines, which are useful ligands and useful reagents.
Kirsanov reaction	Kirsanov reaction	A typical reaction involves triphenylphosphine with bromine to give bromotriphenylphosphonium bromide: Ph3P + Br2 → Ph3PBr+Br−This salt is treated in situ with alkylamines to give the iminophosphorane: Ph3PBr+Br− + 3 RNH2 → Ph3PNR + 2 RNH3+Br−The method is used when the conventional Staudinger reaction is not applicable, i.e. when the organic azide is not available to generate the iminophosphorane. Thus, it is used to make iminophosphoranes from alkyl amines.
Pasta allo scarpariello	Pasta allo scarpariello	Pasta allo scarpariello is a traditional Italian pasta dish from Naples.It is typically made with spaghetti, tomatoes, Pecorino Romano cheese, Parmigiano Reggiano cheese, basil, chili pepper, extra virgin olive oil, garlic and salt.Its name literally means "shoemaker's pasta".
Epigram	Epigram	An epigram is a brief, interesting, memorable, and sometimes surprising or satirical statement. The word is derived from the Greek ἐπίγραμμα epígramma "inscription" from ἐπιγράφειν epigráphein "to write on, to inscribe", and the literary device has been employed for over two millennia. The presence of wit or sarcasm tends to distinguish non-poetic epigrams from aphorisms and adages, which tend to lack those qualities.
Epigram	Ancient Greek	The Greek tradition of epigrams began as poems inscribed on votive offerings at sanctuaries – including statues of athletes – and on funerary monuments, for example "Go tell it to the Spartans, passersby...". These original epigrams did the same job as a short prose text might have done, but in verse. Epigram became a literary genre in the Hellenistic period, probably developing out of scholarly collections of inscriptional epigrams.
Epigram	Ancient Greek	Though modern epigrams are usually thought of as very short, Greek literary epigram was not always as short as later examples, and the divide between "epigram" and "elegy" is sometimes indistinct (they share a characteristic metre, elegiac couplets). In the classical period, the clear distinction between them was that epigrams were inscribed and meant to be read, while elegies were recited and meant to be heard. Some elegies could be quite short, but only public epigrams were longer than ten lines. All the same, the origin of epigram in inscription exerted a residual pressure to keep things concise, even when they were recited in Hellenistic times. Many of the characteristic types of literary epigram look back to inscriptional contexts, particularly funerary epigram, which in the Hellenistic era becomes a literary exercise. Many "sympotic" epigrams combine sympotic and funerary elements – they tell their readers (or listeners) to drink and live for today because life is short. Generally, any theme found in classical elegies could be and were adapted for later literary epigrams.
Epigram	Ancient Greek	Hellenistic epigrams are also thought of as having a "point" – that is, the poem ends in a punchline or satirical twist. By no means do all Greek epigrams behave this way; many are simply descriptive, but Meleager of Gadara and Philippus of Thessalonica, the first comprehensive anthologists, preferred the short and witty epigram. Since their collections helped form knowledge of the genre in Rome and then later throughout Europe, Epigram came to be associated with 'point', especially because the European epigram tradition takes the Latin poet Martial as its principal model; he copied and adapted Greek models (particularly the contemporary poets Lucillius and Nicarchus) selectively and in the process redefined the genre, aligning it with the indigenous Roman tradition of "satura", hexameter satire, as practised by (among others) his contemporary Juvenal. Greek epigram was actually much more diverse, as the Milan Papyrus now indicates.
Epigram	Ancient Greek	A major source for Greek literary epigram is the Greek Anthology, a compilation from the 10th century AD based on older collections, including those of Meleager and Philippus. It contains epigrams ranging from the Hellenistic period through the Imperial period and Late Antiquity into the compiler's own Byzantine era – a thousand years of short elegiac texts on every topic under the sun. The Anthology includes one book of Christian epigrams as well as one book of erotic and amorous homosexual epigrams called the Μοῦσα Παιδικἠ (Mousa Paidike, "The Boyish Muse").
Epigram	Ancient Roman	Roman epigrams owe much to their Greek predecessors and contemporaries. Roman epigrams, however, were often more satirical than Greek ones, and at times used obscene language for effect. Latin epigrams could be composed as inscriptions or graffiti, such as this one from Pompeii, which exists in several versions and seems from its inexact meter to have been composed by a less educated person. Its content makes it clear how popular such poems were: Admiror, O paries, te non cecidisse ruinis qui tot scriptorum taedia sustineas.I'm astonished, wall, that you haven't collapsed into ruins, since you're holding up the weary verse of so many poets.However, in the literary world, epigrams were most often gifts to patrons or entertaining verse to be published, not inscriptions. Many Roman writers seem to have composed epigrams, including Domitius Marsus, whose collection Cicuta (now lost) was named after the poisonous plant Cicuta for its biting wit, and Lucan, more famous for his epic Pharsalia. Authors whose epigrams survive include Catullus, who wrote both invectives and love epigrams – his poem 85 is one of the latter.
Epigram	Ancient Roman	Odi et amo. Quare id faciam fortasse requiris.
Epigram	Ancient Roman	Nescio, sed fieri sentio, et excrucior.I hate and I love. Maybe you'd like to know why I do? I don't know, but I feel it happening, and I am tormented.Martial, however, is considered to be the master of the Latin epigram. His technique relies heavily on the satirical poem with a joke in the last line, thus drawing him closer to the modern idea of epigram as a genre. Here he defines his genre against a (probably fictional) critic (in the latter half of 2.77): Disce quod ignoras: Marsi doctique Pedonis saepe duplex unum pagina tractat opus.
Epigram	Ancient Roman	Non sunt longa quibus nihil est quod demere possis, sed tu, Cosconi, disticha longa facis.Learn what you don't know: one work of (Domitius) Marsus or learned Pedo often stretches out over a doublesided page. A work isn't long if you can't take anything out of it, but you, Cosconius, write even a couplet too long.Poets known for their epigrams whose work has been lost include Cornificia.
Epigram	English	In early English literature the short couplet poem was dominated by the poetic epigram and proverb, especially in the translations of the Bible and the Greek and Roman poets.
Epigram	English	Two successive lines of verse that rhyme with each other are known as a couplet. Since 1600, the couplet has been featured as a part of the longer sonnet form, most notably in William Shakespeare's sonnets. Sonnet 76 is an example. The two-line poetic form as a closed couplet was also used by William Blake in his poem "Auguries of Innocence", and also by Byron in his poem Don Juan, by John Gay in his fables, and by Alexander Pope in his An Essay on Man.
Epigram	English	The first work of English literature penned in North America was Robert Hayman's Quodlibets, Lately Come Over from New Britaniola, Old Newfoundland, which is a collection of over 300 epigrams, many of which do not conform to the two-line rule or trend. While the collection was written between 1618 and 1628 in what is now Harbour Grace, Newfoundland, it was published shortly after his return to Britain.In Victorian times the epigram couplet was often used by the prolific American poet Emily Dickinson. Her poem No. 1534 is a typical example of her eleven poetic epigrams. The novelist George Eliot also included couplets throughout her writings. Her best example is in her sequenced sonnet poem entitled Brother and Sister in which each of the eleven sequenced sonnet ends with a couplet. In her sonnets, the preceding lead-in-line, to the couplet ending of each, could be thought of as a title for the couplet, as is shown in Sonnet VIII of the sequence.
Epigram	English	During the early 20th century, the rhymed epigram couplet form developed into a fixed verse image form, with an integral title as the third line. Adelaide Crapsey codified the couplet form into a two-line rhymed verse of ten syllables per line with her image couplet poem On Seeing Weather-Beaten Trees, first published in 1915. By the 1930s, the five-line cinquain verse form became widely known in the poetry of the Scottish poet William Soutar. These were originally labelled epigrams but later identified as image cinquains in the style of Adelaide Crapsey. J. V. Cunningham was also a noted writer of epigrams (a medium suited to a "short-breathed" person).
Epigram	Poetic epigrams	What is an Epigram? a dwarfish whole, Its body brevity, and wit its soul. — Samuel Taylor Coleridge ("Epigram", 1809)Some can gaze and not be sick But I could never learn the trick. There's this to say for blood and breath; They give a man a taste for death. — A. E. HousmanLittle strokes Fell great oaks. — Benjamin FranklinHere lies my wife: here let her lie! Now she's at rest – and so am I. — John DrydenThree Poets, in three distant Ages born, Greece, Italy, and England did adorn. The First in loftiness of thought surpassed; The Next in Majesty; in both the Last. The force of Nature could no farther go: To make a third she joined the former two. — John Dryden ("Epigram on Milton", 1688 (Epigram about John Milton: many poets commented on Milton, including DrydenWe have a pretty witty king, Whose word no man relies on. He never said a foolish thing, And never did a wise one. — John Wilmot, 2nd Earl of Rochester (epigram about Charles II of England)I am His Highness' dog at Kew; Pray tell me, sir, whose dog are you? — Alexander PopeI'm tired of Love: I'm still more tired of Rhyme. But Money gives me pleasure all the time. — Hilaire BellocI hope for nothing. I fear nothing. I am free. — Nikos KazantzakisTo define the beautiful is to misunderstand it. — Charles Robert Anon (Fernando Pessoa)This Humanist whom no belief constrained Grew so broad-minded he was scatter-brained. — J.V. CunninghamAll things pass Love and mankind is grass. — Stevie Smith
Epigram	In art	"When Guns Speak, Death Settles Disputes" is Charles Marion Russell's epigrammatic title for a clash by gunfighters of the Old West in America.
Goldbach's conjecture	Goldbach's conjecture	Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers. The conjecture has been shown to hold for all integers less than 4×1018, but remains unproven despite considerable effort.
Goldbach's conjecture	History	On 7 June 1742, the German mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII), in which he proposed the following conjecture: Goldbach was following the now-abandoned convention of considering 1 to be a prime number, so that a sum of units would indeed be a sum of primes. He then proposed a second conjecture in the margin of his letter, which implies the first: ... eine jede Zahl, die grösser ist als 2, ein aggregatum trium numerorum primorum sey. Every integer greater than 2 can be written as the sum of three primes. Euler replied in a letter dated 30 June 1742 and reminded Goldbach of an earlier conversation they had had ("... so Ew vormals mit mir communicirt haben ..."), in which Goldbach had remarked that the first of those two conjectures would follow from the statement This is in fact equivalent to his second, marginal conjecture. In the letter dated 30 June 1742, Euler stated: Dass ... ein jeder numerus par eine summa duorum primorum sey, halte ich für ein ganz gewisses theorema, ungeachtet ich dasselbe nicht demonstriren kann.That ... every even integer is a sum of two primes, I regard as a completely certain theorem, although I cannot prove it. Each of the three conjectures above has a natural analog in terms of the modern definition of a prime, under which 1 is excluded.
Goldbach's conjecture	History	A modern version of the first conjecture is: A modern version of the marginal conjecture is: And a modern version of Goldbach's older conjecture of which Euler reminded him is: These modern versions might not be entirely equivalent to the corresponding original statements. For example, if there were an even integer N = p + 1 larger than 4, for p a prime, that could not be expressed as the sum of two primes in the modern sense, then it would be a counterexample to the modern version of the third conjecture (without being a counterexample to the original version). The modern version is thus probably stronger (but in order to confirm that, one would have to prove that the first version, freely applied to any positive even integer n, could not possibly rule out the existence of such a specific counterexample N). In any case, the modern statements have the same relationships with each other as the older statements did. That is, the second and third modern statements are equivalent, and either implies the first modern statement.
Goldbach's conjecture	History	The third modern statement (equivalent to the second) is the form in which the conjecture is usually expressed today. It is also known as the "strong", "even", or "binary" Goldbach conjecture. A weaker form of the second modern statement, known as "Goldbach's weak conjecture", the "odd Goldbach conjecture", or the "ternary Goldbach conjecture", asserts that A proof for the weak conjecture was proposed in 2013 by Harald Helfgott. Helfgott's proof has not yet appeared in a peer-reviewed publication, though was accepted for publication in the Annals of Mathematics Studies series in 2015 and has been undergoing further review and revision since. The weak conjecture would be a corollary of the strong conjecture: if n − 3 is a sum of two primes, then n is a sum of three primes. However, the converse implication and thus the strong Goldbach conjecture remain unproven.
Goldbach's conjecture	Verified results	For small values of n, the strong Goldbach conjecture (and hence the weak Goldbach conjecture) can be verified directly. For instance, in 1938, Nils Pipping laboriously verified the conjecture up to n = 100000. With the advent of computers, many more values of n have been checked; T. Oliveira e Silva ran a distributed computer search that has verified the conjecture for n ≤ 4×1018 (and double-checked up to 4×1017) as of 2013. One record from this search is that 3325581707333960528 is the smallest number that cannot be written as a sum of two primes where one is smaller than 9781.
Goldbach's conjecture	Heuristic justification	Statistical considerations that focus on the probabilistic distribution of prime numbers present informal evidence in favour of the conjecture (in both the weak and strong forms) for sufficiently large integers: the greater the integer, the more ways there are available for that number to be represented as the sum of two or three other numbers, and the more "likely" it becomes that at least one of these representations consists entirely of primes.
Goldbach's conjecture	Heuristic justification	A very crude version of the heuristic probabilistic argument (for the strong form of the Goldbach conjecture) is as follows. The prime number theorem asserts that an integer m selected at random has roughly a 1/ln m chance of being prime. Thus if n is a large even integer and m is a number between 3 and n/2, then one might expect the probability of m and n − m simultaneously being prime to be 1/ln m ln(n − m). If one pursues this heuristic, one might expect the total number of ways to write a large even integer n as the sum of two odd primes to be roughly ln ln ln ⁡n)2.
Goldbach's conjecture	Heuristic justification	Since ln n ≪ √n, this quantity goes to infinity as n increases, and one would expect that every large even integer has not just one representation as the sum of two primes, but in fact very many such representations.
Goldbach's conjecture	Heuristic justification	This heuristic argument is actually somewhat inaccurate, because it assumes that the events of m and n − m being prime are statistically independent of each other. For instance, if m is odd, then n − m is also odd, and if m is even, then n − m is even, a non-trivial relation because, besides the number 2, only odd numbers can be prime. Similarly, if n is divisible by 3, and m was already a prime other than 3, then n − m would also be coprime to 3 and thus be slightly more likely to be prime than a general number. Pursuing this type of analysis more carefully, G. H. Hardy and John Edensor Littlewood in 1923 conjectured (as part of their Hardy–Littlewood prime tuple conjecture) that for any fixed c ≥ 2, the number of representations of a large integer n as the sum of c primes n = p1 + ⋯ + pc with p1 ≤ ⋯ ≤ pc should be asymptotically equal to ln ln ⁡xc, where the product is over all primes p, and γc,p(n) is the number of solutions to the equation n = q1 + ⋯ + qc mod p in modular arithmetic, subject to the constraints q1, …, qc ≠ 0 mod p. This formula has been rigorously proven to be asymptotically valid for c ≥ 3 from the work of Ivan Matveevich Vinogradov, but is still only a conjecture when c = 2. In the latter case, the above formula simplifies to 0 when n is odd, and to ln ln ⁡n)2 when n is even, where Π2 is Hardy–Littlewood's twin prime constant := 0.66016 18158 46869 57392 78121 10014 … This is sometimes known as the extended Goldbach conjecture. The strong Goldbach conjecture is in fact very similar to the twin prime conjecture, and the two conjectures are believed to be of roughly comparable difficulty.
Goldbach's conjecture	Heuristic justification	The Goldbach partition function is the function that associates to each even integer the number of ways it can be decomposed into a sum of two primes. Its graph looks as a comet, and is therefore called Goldbach's comet.Goldbach's comet suggests tight upper and lower bounds on the number of representations of an even number as the sum of two primes, and also that the number of these representations depend strongly on the value modulo 3 of the number.
Goldbach's conjecture	Rigorous results	The strong Goldbach conjecture is much more difficult than the weak Goldbach conjecture. Using Vinogradov's method, Nikolai Chudakov, Johannes van der Corput, and Theodor Estermann showed that almost all even numbers can be written as the sum of two primes (in the sense that the fraction of even numbers up to some N which can be so written tends towards 1 as N increases). In 1930, Lev Schnirelmann proved that any natural number greater than 1 can be written as the sum of not more than C prime numbers, where C is an effectively computable constant; see Schnirelmann density. Schnirelmann's constant is the lowest number C with this property. Schnirelmann himself obtained C < 800000. This result was subsequently enhanced by many authors, such as Olivier Ramaré, who in 1995 showed that every even number n ≥ 4 is in fact the sum of at most 6 primes. The best known result currently stems from the proof of the weak Goldbach conjecture by Harald Helfgott, which directly implies that every even number n ≥ 4 is the sum of at most 4 primes.In 1924, Hardy and Littlewood showed under the assumption of the generalized Riemann hypothesis that the number of even numbers up to X violating the Goldbach conjecture is much less than X1⁄2 + c for small c.In 1948, using sieve theory, Alfréd Rényi showed that every sufficiently large even number can be written as the sum of a prime and an almost prime with at most K factors. Chen Jingrun showed in 1973 using the methods of sieve theory that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime (the product of two primes). See Chen's theorem for further information.
Goldbach's conjecture	Rigorous results	In 1975, Hugh Montgomery and Robert Charles Vaughan showed that "most" even numbers are expressible as the sum of two primes. More precisely, they showed that there exist positive constants c and C such that for all sufficiently large numbers N, every even number less than N is the sum of two primes, with at most CN1 − c exceptions. In particular, the set of even integers that are not the sum of two primes has density zero.
Goldbach's conjecture	Rigorous results	In 1951, Yuri Linnik proved the existence of a constant K such that every sufficiently large even number is the sum of two primes and at most K powers of 2. János Pintz and Imre Ruzsa found in 2020 that K = 8 works. Assuming the generalized Riemann hypothesis, K = 7 also works, as shown by Roger Heath-Brown and Jan-Christoph Schlage-Puchta in 2002.
Goldbach's conjecture	Related problems	Although Goldbach's conjecture implies that every positive integer greater than one can be written as a sum of at most three primes, it is not always possible to find such a sum using a greedy algorithm that uses the largest possible prime at each step. The Pillai sequence tracks the numbers requiring the largest number of primes in their greedy representations.Similar problems to Goldbach's conjecture exist in which primes are replaced by other particular sets of numbers, such as the squares: It was proven by Lagrange that every positive integer is the sum of four squares. See Waring's problem and the related Waring–Goldbach problem on sums of powers of primes.
Goldbach's conjecture	Related problems	Hardy and Littlewood listed as their Conjecture I: "Every large odd number (n > 5) is the sum of a prime and the double of a prime". This conjecture is known as Lemoine's conjecture and is also called Levy's conjecture. The Goldbach conjecture for practical numbers, a prime-like sequence of integers, was stated by Margenstern in 1984, and proved by Melfi in 1996: every even number is a sum of two practical numbers.
Goldbach's conjecture	Related problems	A strengthening of the Goldbach conjecture proposed by Harvey Dubner states that every even integer greater than 4208 is the sum of two twin primes. Only 34 even integers less than 4208 are not the sum of two twin primes. Dubner has verified computationally that this list is complete up to 2×1010. A proof of this stronger conjecture would not only imply Goldbach's conjecture, but also the twin prime conjecture.
Goldbach's conjecture	In popular culture	Goldbach's Conjecture (Chinese: 哥德巴赫猜想) is the title of the biography of Chinese mathematician and number theorist Chen Jingrun, written by Xu Chi.
Goldbach's conjecture	In popular culture	The conjecture is a central point in the plot of the 1992 novel Uncle Petros and Goldbach's Conjecture by Greek author Apostolos Doxiadis, in the short story "Sixty Million Trillion Combinations" by Isaac Asimov and also in the 2008 mystery novel No One You Know by Michelle Richmond.Goldbach's conjecture is part of the plot of the 2007 Spanish film Fermat's Room.
Voluntary Control Council for Interference by Information Technology Equipment	Voluntary Control Council for Interference by Information Technology Equipment	The Voluntary Control Council for Interference by Information Technology Equipment or VCCI is the Japanese body governing RF emissions (i.e. electromagnetic interference) standards. [1] It was formed in December 1985. The VCCI mark of conformance also appears on some electrical equipment sold outside Japan.
Ski cross	Ski cross	Ski cross is a skiing competition which incorporates terrain features traditionally found in freestyle skiing with courses which include big-air jumps and high-banked turns. In spite of the fact that it is a timed racing event, it is often considered a type of freestyle skiing. What sets ski cross apart from other alpine skiing disciplines is that it involves more than one skier racing down the course. Any intentional contact with other competitors like grabbing or any other forms of contact meant to give the competitor an advantage leads to disqualification.
Ski cross	Ski cross	Ski cross is a part of the FIS Freestyle World Ski Championships, the world championship organized by the FIS for freestyle skiing. First organized in 1986, the world championship is now held every odd year. In 2010 the sport debuted as a part of the Winter Olympic Games and has been contested ever since. It was a part of the Winter X Games until 2012.
Ski cross	Overview	In a time trial or qualification round, every competitor skis down the course, which is built to encompass both naturally occurring terrain and artificial features like jumps, rollers or banks. After the time trial, the fastest 32 skiers (fastest 16 if not 32 competitors) compete in a knockout series in rounds of four. A group of four skiers start simultaneously and attempt to reach the end of the course. The first two to cross the finish line will advance to the next round. At the end, the big final and small final rounds determine 1st to 4th and 5th to 8th places, respectively.
Ski cross	History	The idea for a multi-racer single run with obstacles seems to have been borne at Alyeska Ski Resort in Alaska (USA) during the late 1970s. A group of racers, led by Scott Hunter an employee at Alyeska wanted to take advantage of the mountain's natural bobsled-like gullies and rollers in a race that was a hybrid between a downhill ski race and Motocross. It eventually evolved into a race using up to 5 skiers on the course at the same time, all racing against each other. No ski poles were allowed, and racers were allowed to interfere and contact other racers as much as they wished. As a result, there were typically several falls (and some injuries) from intentional collisions during each run. Interest waned in the early 1980s due to athletes graduating high school and leaving for college, while other racers concentrated on USSA and FIS sanctioned events. The last ski cross event on the original “silvertip” track occurred in the early 1980s.
Ski cross	History	A similar idea originated with Jim "Too Tall" Essick, one of the founders of Recreational Sports Marketing (RSM), in the late 1980s. Essick wanted to bring the excitement of motocross to skiing, in order to make ski races more exciting for spectators. The idea was pitched to several corporations, but none wanted to sponsor the concept at the time. In 1991, a television programme filmed a snowboard cross segment, and the name "boarder cross" was trademarked. Eventually, similar events were staged with skis and, thus, skier cross was born.
Ski cross	FIS Freestyle World Ski Championships	In addition to moguls and aerials, ski cross competitions were added to the International Ski Federation (FIS)'s FIS Freestyle Skiing World Cup calendar in 2004.
Ski cross	Winter Olympic Games	Ski cross debuted in the Olympics at the 2010 Winter Olympics where Michael Schmid won the men's event, and Ashleigh McIvor of Canada won the women's event. In the 2014 Winter Olympics France's Men swept the podium while in the women's event, Canadians Marielle Thompson and Kelsey Serwa finished first and second respectively. Swedish athlete Anna Holmlund took bronze.
Ski cross	Winter Olympic Games	In the 2018 Winter Olympics in Korea, Canada continued its domination of the sport. Kelsey Serwa won her second Olympic medal, this time a gold. Canadian teammate Brittany Phelan took home the silver. Swiss skier Fanny Smith won bronze. On the men's side, Brady Leman got redemption after crashing in the final at Sochi by winning gold in Korea. Swiss athlete Marc Bischofberger won silver and Russian Sergey Ridzik won bronze (competing under the Olympic Flag).
Ski cross	Winter X Games	Ski cross was in the first fifteen Winter X Games, an event which features extreme sports, and was in all Winter X Games until the 2012 Winter X Games. Ski cross, boardercross, and mono ski cross were cut from the 2013 Winter X Games due to the cost of building the cross course.

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