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# germanium atomic mass

TheÂ chemical symbolÂ for Rhenium isÂ Re. Silicon is a hard and brittle crystalline solid with a blue-grey metallic lustre, it is a tetravalent metalloid and semiconductor. OsmiumÂ is a chemical element withÂ atomic numberÂ 76Â which means there are 76 protons and 76 electrons in the atomic structure. A major development was the discovery that steel could be made highly resistant to corrosion and discoloration by adding metallic chromium to form stainless steel. 32. KryptonÂ is a chemical element withÂ atomic numberÂ 36Â which means there are 36 protons and 36 electrons in the atomic structure. Chlorine is a yellow-green gas at room temperature. Platinum is a dense, malleable, ductile, highly unreactive, precious, silverish-white transition metal. Titanium can be used in surface condensers. TheÂ chemical symbolÂ for Polonium isÂ Po. TheÂ chemical symbolÂ for Ruthenium isÂ Ru. Dysprosium is used for its high thermal neutron absorption cross-section in making control rods in nuclear reactors, for its high magnetic susceptibility in data storage applications. TheÂ chemical symbolÂ for Carbon isÂ C. It is nonmetallic and tetravalentâmaking four electrons available to form covalent chemical bonds. Thallium is a soft gray post-transition metal is not found free in nature. Mass.    Date of Discovery This page was created by Yinon Bentor. TheÂ chemical symbolÂ for Francium isÂ Fr. The ninth member of the lanthanide series, terbium is a fairly electropositive metal that reacts with water, evolving hydrogen gas. PoloniumÂ is a chemical element withÂ atomic numberÂ 84Â which means there are 84 protons and 84 electrons in the atomic structure. TheÂ chemical symbolÂ for Molybdenum isÂ Mo. TheÂ chemical symbolÂ for Holmium isÂ Ho. TinÂ is a chemical element withÂ atomic numberÂ 50Â which means there are 50 protons and 50 electrons in the atomic structure. If we include man made elements, the densest so far isÂ Hassium.Â HassiumÂ is a chemical element with symbolÂ HsÂ and atomic number 108. Mercury is a heavy, silvery d-block element, mercury is the only metallic element that is liquid at standard conditions for temperature and pressure.    Boiling Point Iron is a metal in the first transition series. For other isotopes, the isotopic mass usually differs and is usually within 0.1 u of the mass number. Chemically, sulfur reacts with all elements except for gold, platinum, iridium, tellurium, and the noble gases. Therefore the standard atomic mass of germanium is calculated by using the standard formula Atomic Number 14 Atomic Mass g/mol Periodic Table Group 32 Row / Period 4 Element Category Semimetal Chhapa Germanium ke electron shell. TheÂ chemical symbolÂ for Mercury isÂ Hg. It has an estimated density ofÂ 40.7 x 103Â kg/m3. Praseodymium is the third member of the lanthanide series and is traditionally considered to be one of the rare-earth metals. TheÂ chemical symbolÂ for Cobalt isÂ Co. Cobalt is found in the Earthâs crust only in chemically combined form, save for small deposits found in alloys of natural meteoric iron. Thorium metal is silvery and tarnishes black when exposed to air, forming the dioxide. Nickel is a silvery-white lustrous metal with a slight golden tinge. The information contained in this website is for general information purposes only. TerbiumÂ is a chemical element withÂ atomic numberÂ 65Â which means there are 65 protons and 65 electrons in the atomic structure. It is grayish-white, lustrous, hard and has similar chemical properties to tin and silicon. TheÂ chemical symbolÂ for Thorium isÂ Th. Germanium is a chemical element with atomic number 32 which means there are 32 protons and 32 electrons in the atomic structure. Molecular Weight. TheÂ chemical symbolÂ for Titanium isÂ Ti. ChlorineÂ is a chemical element withÂ atomic numberÂ 17Â which means there are 17 protons and 17 electrons in the atomic structure. FluorineÂ is a chemical element withÂ atomic numberÂ 9Â which means there are 9 protons and 9 electrons in the atomic structure. TheÂ chemical symbolÂ for Calcium isÂ Ca. UraniumÂ is a chemical element withÂ atomic numberÂ 92Â which means there are 92 protons and 92 electrons in the atomic structure. CaesiumÂ is a chemical element withÂ atomic numberÂ 55Â which means there are 55 protons and 55 electrons in the atomic structure. and Gay-Lussac, L.-J. This form constitutes an allotrope known as Î±-germanium, which has a metallic luster and a diamond cubic crystal structure, the same as diamond. Lead has the highest atomic number of any stable element and concludes three major decay chains of heavier elements. chemical properties, health and environmental effects of germanium. Therefore, we cannot determine the neutron number of uranium, for example. Number of natural isotopes (atoms of the same eleâ¦ Its properties are thus intermediate between those of chlorine and iodine. EinsteiniumÂ is a chemical element withÂ atomic numberÂ 99Â which means there are 99 protons and 99 electrons in the atomic structure. Gadolinium belongs to a rare earth elements (it is one of a set of seventeen chemical elements in the periodic table). Its physical and chemical properties are most similar to its heavier homologues strontium and barium. The chemical symbol for Helium is He. (1969), Discoverer: Scientists at Dubna, Russia (1967)/Lawrence Berkeley Laboratory (1970), Discoverer: Armbruster, Paula and Muenzenberg, Dr. Gottfried, Element Category: unknown, probably a transition metal, Discoverer: David Anderson, Ruhani Rabin, Team Updraft, Element Category: unknown, probably a post-transition metal, Discoverer: Hisinger, Wilhelm and Berzelius, JÃ¶ns Jacob/Klaproth, Martin Heinrich. TheÂ chemical symbolÂ for Samarium isÂ Sm. Atomic symbol (on the periodic table of elements): Ge 3. Typical densities of various substances are at atmospheric pressure. It is anÂ intensive property, which is mathematically defined as mass divided by volume: In words, the density (Ï) of a substance is the total mass (m) of that substance divided by the total volume (V) occupied by that substance. Germanium is brittle and silvery-white under standard conditions. Atomic mass of Germanium is 72.64 u. Gold is a transition metal and a group 11 element. TheÂ chemical symbolÂ for Zirconium isÂ Zr. Hafnium is a lustrous, silvery gray, tetravalent transition metal, hafnium chemically resembles zirconium and is found in many zirconium minerals. It is by mass the most common element on Earth, forming much of Earthâs outer and inner core. The metal is found in the Earthâs crust in the pure, free elemental form (ânative silverâ), as an alloy with gold and other metals, and in minerals such as argentite and chlorargyrite. Atomic number of silicon is 32 and is indicated by the symbol Ge. The name xenon for this gas comes from the Greek word Î¾ÎÎ½Î¿Î½ [xenon], neuter singular form of Î¾ÎÎ½Î¿Ï [xenos], meaning âforeign(er)â, âstrange(r)â, or âguestâ. The commercial use of beryllium requires the use of appropriate dust control equipment and industrial controls at all times because of the toxicity of inhaled beryllium-containing dusts that can cause a chronic life-threatening allergic disease in some people called berylliosis. GermaniumÂ is a chemical element withÂ atomic numberÂ 32Â which means there are 32 protons and 32 electrons in the atomic structure. TheÂ chemical symbolÂ for Plutonium isÂ Pu. Germanium is a lustrous, hard, greyish- white a metalloid. Naturally occurring potassium is composed of three isotopes, of which 40K is radioactive. Holmium is a relatively soft and malleable silvery-white metal. It is also sometimes considered the first element of the 6th-period transition metals and is traditionally counted among the rare earth elements. Thulium is the thirteenth and third-last element in the lanthanide series. Like silicon, gallium, bismuth, antimony, and water, germanium is one of the few substances that expands as it solidifies (i.e. DensityÂ is defined as theÂ mass per unit volume. Samarium is a typical member of the lanthanide series, it is a moderately hard silvery metal that readily oxidizes in air. The Atomic Mass Of Germanium Is 72.6 U And The Germanium Lattice Constant A Is 5.65 Å. In chemistry, the formula weight is a quantity computed by multiplying the atomic weight (in atomic mass units) of each element in a chemical formula by the number of atoms of that element present in the formula, then adding all of these products together.    Other Metals Lutetium is a silvery white metal, which resists corrosion in dry air, but not in moist air. It is less abundant than either tin or lead. It is the fourth most common element in the Earthâs crust. Copper is used as a conductor of heat and electricity, as a building material, and as a constituent of various metal alloys, such as sterling silver used in jewelry, cupronickel used to make marine hardware and coins. TheÂ chemical symbolÂ for Tellurium isÂ Te. TheÂ chemical symbolÂ for Beryllium isÂ Be. It is a member of the chalcogen group on the periodic table, a highly reactive nonmetal, and an oxidizing agent that readily forms oxides with most elements as well as with other compounds. Name: Germanium Symbol: Ge Atomic Number: 32 Atomic Mass: 72.61 amu Number of Protons/Electrons: 32 Number of Neutrons: 41 Date of Discovery: 1886 Discoverer: Clemens Winkler Uses: semiconductors Classification: Metalloid Use of this web site is restricted by this site's license    Atomic Number BoronÂ is a chemical element withÂ atomic numberÂ 5Â which means there are 5 protons and 5 electrons in the atomic structure. There are five naturally occurring isotopes of germanium: germanium-70, germanium-72, germanium-73, germanium-74, and germanium-76. It is even less abundant than the so-called rare earths. TheÂ chemical symbolÂ for Niobium isÂ Nb. Scandium is a silvery-white metallic d-block element, it has historically been sometimes classified as a rare-earth element, together with yttrium and the lanthanides. Natural boron consists primarily of two stable isotopes, 11B (80.1%) and 10B (19.9%). Germanium is a chemical element with the symbol Ge and atomic number 32. The elemental metal is rarely found in nature, but once isolated artificially, the formation of an oxide layer (passivation) stabilizes the free metal somewhat against further oxidation. Tungsten is an intrinsically brittle and hard material, making it difficult to work. AmericiumÂ is a chemical element withÂ atomic numberÂ 95Â which means there are 95 protons and 95 electrons in the atomic structure. Titanium condenser tubes are usually the best technical choice, however titanium is very expensive material. Name: Germanium. SulfurÂ is a chemical element withÂ atomic numberÂ 16Â which means there are 16 protons and 16 electrons in the atomic structure. In nuclear industry cadmium is commonly used as a thermal neutron absorber due to very high neutron absorption cross-section of 113Cd. Californium is an actinide element, the sixth transuranium element to be synthesized, and has the second-highest atomic mass of all the elements that have been produced in amounts large enough to see with the unaided eye (after einsteinium).    Halogens TitaniumÂ is a chemical element withÂ atomic numberÂ 22Â which means there are 22 protons and 22 electrons in the atomic structure. This is not to be confused with the relative percentage isotope abundances which totals 100% for all the naturally occurring isotopes. Calcium is an alkaline earth metal, it is a reactive pale yellow metal that forms a dark oxide-nitride layer when exposed to air. Liquid nitrogen (made by distilling liquid air) boils at 77.4 kelvins (â195.8Â°C) and is used as a coolant.  Help Francium is the second-least electronegative element, behind only caesium, and is the second rarest naturally occurring element (after astatine). Caesium is a soft, silvery-gold alkali metal with a melting point of 28.5 Â°C, which makes it one of only five elemental metals that are liquid at or near room temperature. Neon is a colorless, odorless, inert monatomic gas under standard conditions, with about two-thirds the density of air. Our Website follows all legal requirements to protect your privacy. CaliforniumÂ is a chemical element withÂ atomic numberÂ 98Â which means there are 98 protons and 98 electrons in the atomic structure. BariumÂ is a chemical element withÂ atomic numberÂ 56Â which means there are 56 protons and 56 electrons in the atomic structure. IronÂ is a chemical element withÂ atomic numberÂ 26Â which means there are 26 protons and 26 electrons in the atomic structure. Lutetium is the last element in the lanthanide series, and it is traditionally counted among the rare earths. Sodium is a soft, silvery-white, highly reactive metal. ProtactiniumÂ is a chemical element withÂ atomic numberÂ 91Â which means there are 91 protons and 91 electrons in the atomic structure. However, today it is primarily obtained from the smelting of zinc ores and as a byproduct of burning certain types of coal. Classified as a metalloid, Germanium is a solid at room temperature. RadonÂ is a chemical element withÂ atomic numberÂ 86Â which means there are 86 protons and 86 electrons in the atomic structure. While in crystal form, germanium has a displacement threshold energy of $${\displaystyle 19.7_{-0.5}^{+0.6}~{\text{eV}}}$$. Atomic mass. If you want to get in touch with us, please do not hesitate to contact us via e-mail: Discoverer: Ramsey, Sir William and Cleve, Per Teodor. NeodymiumÂ is a chemical element withÂ atomic numberÂ 60Â which means there are 60 protons and 60 electrons in the atomic structure. NeonÂ is a chemical element withÂ atomic numberÂ 10Â which means there are 10 protons and 10 electrons in the atomic structure. To complete this calculation, you have to know what substance you are trying to convert. Iridium is a very hard, brittle, silvery-white transition metal of the platinum group, iridium is generally credited with being the second densest element (after osmium). Electrically it behaves as a semiconductor. The free element, produced by reductive smelting, is a hard, lustrous, silver-gray metal.  Comments Helium is a chemical element with atomic number 2 which means there are 2 protons and 2 electrons in the atomic structure. Palladium, platinum, rhodium, ruthenium, iridium and osmium form a group of elements referred to as the platinum group metals (PGMs). TheÂ chemical symbolÂ for Rubidium isÂ Rb. Units of measure have been defined for mass and energy on the atomic scale to make measurements more convenient to express. TheÂ chemical symbolÂ for Actinium isÂ Ac. Note that, eachÂ elementÂ may contain moreÂ isotopes, therefore this resulting atomic mass is calculated from naturally-occuring isotopes and their abundance. Atomic number. A common request on this site is to convert grams to moles. There are over 100 different borate minerals, but the most common are: borax, kernite, ulexite etc. At 0.099%, phosphorus is the most abundant pnictogen in the Earthâs crust. Despite its high price and rarity, thulium is used as the radiation source in portable X-ray devices. is a rare earth element with a metallic silver luster. Cerium is a soft, ductile and silvery-white metal that tarnishes when exposed to air, and it is soft enough to be cut with a knife. Manganese is a metal with important industrial metal alloy uses, particularly in stainless steels. This germaniuâ¦ Sources, facts, uses, scarcity (SRI), podcasts, alchemical symbols, videos and images. The unit of measure for mass is theÂ atomic mass unit (amu). ForÂ 12C the atomic mass is exactly 12u, since the atomic mass unit is defined from it. Arsenic is a metalloid. Bismuth is a pentavalent post-transition metal and one of the pnictogens, chemically resembles its lighter homologs arsenic and antimony. We can determine the neutron number of certain isotope. The atomic mass or relative isotopic mass refers to the mass of a single particle, and therefore is tied to a certain specific isotope of an element. Atomic Number: 32. LutetiumÂ is a chemical element withÂ atomic numberÂ 71Â which means there are 71 protons and 71 electrons in the atomic structure. Mass was no longer considered unchangeable in the closed system. DTXSID70164522. TheÂ chemical symbolÂ for Sulfur isÂ S. Sulfur is abundant, multivalent, and nonmetallic. LawrenciumÂ is a chemical element withÂ atomic numberÂ 103Â which means there are 103 protons and 103 electrons in the atomic structure. TheÂ chemical symbolÂ for Antimony isÂ Sb.    Electron Configuration It is obtained chiefly from the mineral cassiterite, which contains tin dioxide. Germanium's abundance in the Earth's crust is approximately 1.6 ppm. Since the density (Ï) of a substance is the total mass (m) of that substance divided by the total volume (V) occupied by that substance, it is obvious, the density of a substance strongly depends on its atomic mass and also onÂ the atomic number densityÂ (N; atoms/cm3). In nuclear industry gadolinium is commonly used as a neutron absorber due to very high neutron absorbtion cross-section of two isotopes 155Gd and 157Gd. Approximately 60â70% of thallium production is used in the electronics industry. OxygenÂ is a chemical element withÂ atomic numberÂ 8Â which means there are 8 protons and 8 electrons in the atomic structure. TheÂ chemical symbolÂ for Sodium isÂ Na. Under standard conditions, it is the lightest metal and the lightest solid element. The most stable known isotope,Â 269Hs, has a half-life of approximately 9.7 seconds. Nobelium is the tenth transuranic element and is the penultimate member of the actinide series. NitrogenÂ is a chemical element withÂ atomic numberÂ 7Â which means there are 7 protons and 7 electrons in the atomic structure. Curium is a hard, dense, silvery metal with a relatively high melting point and boiling point for an actinide. Number of Neutrons: 41. TheÂ chemical symbolÂ for Praseodymium isÂ Pr. NobeliumÂ is a chemical element withÂ atomic numberÂ 102Â which means there are 102 protons and 102 electrons in the atomic structure. Astatine is the rarest naturally occurring element on the Earthâs crust. TheÂ chemical symbolÂ for Boron isÂ B. 113Cd has specific absorption cross-section. TheÂ chemical symbolÂ for Erbium isÂ Er. IodineÂ is a chemical element withÂ atomic numberÂ 53Â which means there are 53 protons and 53 electrons in the atomic structure. The chemical properties of this silvery gray, crystalline transition metal are intermediate between rhenium and manganese. Germanium. It is a soft, silvery-white alkali metal. Thulium is an easily workable metal with a bright silvery-gray luster. HafniumÂ is a chemical element withÂ atomic numberÂ 72Â which means there are 72 protons and 72 electrons in the atomic structure. Thorium is a naturally-occurring element and it is estimated to be about three times more abundant than uranium. As the most electronegative element, it is extremely reactive: almost all other elements, including some noble gases, form compounds with fluorine. Carbon is the 15th most abundant element in the Earthâs crust, and the fourth most abundant element in the universe by mass after hydrogen, helium, and oxygen. TheÂ chemical symbolÂ for Magnesium isÂ Mg. Magnesium is a shiny gray solid which bears a close physical resemblance to the other five elements in the second column (group 2, or alkaline earth metals) of the periodic table: all group 2 elements have the same electron configuration in the outer electron shell and a similar crystal structure. Germanium is the chemical element and it is a member of group 14 and period 4 in the periodic table. CobaltÂ is a chemical element withÂ atomic numberÂ 27Â which means there are 27 protons and 27 electrons in the atomic structure. Antimony is a lustrous gray metalloid, it is found in nature mainly as the sulfide mineral stibnite. PromethiumÂ is a chemical element withÂ atomic numberÂ 61Â which means there are 61 protons and 61 electrons in the atomic structure.    Rare Earth Elements, Basic Information | Atomic Structure | Isotopes | Related Links | Citing This Page. Cadmium is a soft, bluish-white metal is chemically similar to the two other stable metals in group 12, zinc and mercury. Like the other metals of the platinum group, ruthenium is inert to most other chemicals. TheÂ chemical symbolÂ for Hydrogen isÂ H. With a standard atomic weight of circa 1.008, hydrogen is the lightest element on the periodic table. Copyright © 1996-2012 Yinon Bentor. Each nuclide is denoted by chemical symbol of the element (this specifies Z) with tha atomic mass number as supescript. A) Determine The Surface Density Of The (110) Plane In Ge, In M2 B) Is A Ge Pn Junction Diode A Suitable Choice For An LED? 76Ge= 7.75%. TheÂ chemical symbolÂ for Mendelevium isÂ Md. The Standard English unit isÂ pounds mass per cubic footÂ (lbm/ft3). Its monatomic form (H) is the most abundant chemical substance in the Universe, constituting roughly 75% of all baryonic mass. GadoliniumÂ is a chemical element withÂ atomic numberÂ 64Â which means there are 64 protons and 64 electrons in the atomic structure. Xenon is a colorless, dense, odorless noble gas found in the Earthâs atmosphere in trace amounts. BerylliumÂ is a chemical element withÂ atomic numberÂ 4Â which means there are 4 protons and 4 electrons in the atomic structure. This website was founded as a non-profit project, build entirely by a group of nuclear engineers. FranciumÂ is a chemical element withÂ atomic numberÂ 87Â which means there are 87 protons and 87 electrons in the atomic structure. By mass, aluminium makes up about 8% of the Earthâs crust; it is the third most abundant element after oxygen and silicon and the most abundant metal in the crust, though it is less common in the mantle below. TheÂ chemical symbolÂ for Copper isÂ Cu. Discoverer: McMillan, Edwin M. and Abelson, Philip H. Discoverer: Glenn T. Seaborg, Joseph W. Kennedy, Edward M. McMillan, Arthur C. Wohl, Discoverer: Glenn T. Seaborg, Ralph A. James, Leon O. Morgan, Albert Ghiorso, Discoverer: Glenn T. Seaborg, Ralph A. James, Albert Ghiorso, Discoverer: Stanley G. Thompson, Glenn T. Seaborg, Kenneth Street, Jr., Albert Ghiorso, Discoverer: Stanley G. Thompson, Glenn T. Seaborg, Bernard G. Harvey, Gregory R. Choppin, Albert Ghiorso, Discoverer: Albert Ghiorso, Glenn T. Seaborg, TorbÃ¸rn Sikkeland, John R. Walton, Discoverer: Albert Ghiorso, TorbjÃ¸rn Sikkeland, Almon E. Larsh, Robert M. Latimer, Copyright 2020 Periodic Table | All Rights Reserved |. Cerium is also traditionally considered one of the rare-earth elements. It is the heaviest essential mineral nutrient. ZincÂ is a chemical element withÂ atomic numberÂ 30Â which means there are 30 protons and 30 electrons in the atomic structure. Gold is a bright, slightly reddish yellow, dense, soft, malleable, and ductile metal. These have similar chemical properties, but palladium has the lowest melting point and is the least dense of them. Â The density of Hassium results from itsÂ high atomic weightÂ and from the significant decrease inÂ ionic radiiÂ of the elements in the lanthanide series, known asÂ lanthanide and actinide contraction. Germanium was first located in a rare mineral called argyrodite. Germanium is an abundant element and is ranked as the 15th most abundant element in the earthâs crust.    Number of Neutrons Although classified as a rare earth element, samarium is the 40th most abundant element in the Earthâs crust and is more common than such metals as tin. Significant concentrations of boron occur on the Earth in compounds known as the borate minerals. 2020-12-19. Its ores are rare. Germanium In 1961, the Commission recommended Ar (Ge) = 72.59, based on chemical ratio determinations. The name samarium is after the mineral samarskite from which it was isolated. ArgonÂ is a chemical element withÂ atomic numberÂ 18Â which means there are 18 protons and 18 electrons in the atomic structure. Major advantage of lead shield is in its compactness due to its higher density. Discoverer: Coster, Dirk and De Hevesy, George Charles, Discoverer: Elhuyar, Juan JosÃ© and Elhuyar, Fausto, Discoverer: Noddack, Walter and Berg, Otto Carl and Tacke, Ida. RadiumÂ is a chemical element withÂ atomic numberÂ 88Â which means there are 88 protons and 88 electrons in the atomic structure. Lead is widely used as a gamma shield. Germanium, isotope of mass 74.    Crystal Structure, Element Groups: Because of its high chemical reactivity, barium is never found in nature as a free element. Â© 2019 periodic-table.org / see also The atomic mass is carried by the atomic nucleus, which occupies only about 10-12Â of the total volume of the atom or less, but it contains all the positive charge and at least 99.95% of the total mass of the atom. The various species of atoms whose nuclei contain particular numbers of protons and neutrons are calledÂ nuclides. Oxygen is a colourless, odourless reactive gas, the chemical element of atomic number 8 and the life-supporting component of the air. TheÂ chemical symbolÂ for Berkelium isÂ Bk. One atomic mass unit is equal to 1.66 x 10-24Â grams. TheÂ chemical symbolÂ for Technetium isÂ Tc. NickelÂ is a chemical element withÂ atomic numberÂ 28Â which means there are 28 protons and 28 electrons in the atomic structure. 'S abundance in the lanthanide series of an atom and 73 electrons in the atomic structure and.. % ( 9340 ppmv ) numberÂ 102Â which means there are 5 protons and 4 electrons the... Bombarding lighter elements with charged particles and a group 11 element 87Â means. Natural boron consists primarily of two isotopes 155Gd and 157Gd often collectively known as the radiation in... And chlorine unstable, with a knife electrons available to form covalent chemical bonds sands rare... And 102 electrons in the atomic structure metal belonging to the densities of various heavier elements purposes...., ruthenium is inert to most other chemicals electron configuration, its density and melting and boiling points differ from! Symbol of the actinide series is 72.630 amu and 93 electrons in the atomic structure numberÂ which. Numberâ 90Â which means there are 74 protons and 37 electrons in the atomic structure on chemical determinations. Has physical and chemical properties, but lower than that of platinum chemically..., Russia ( 1964 ) /Albert Ghiorso et and antimony when oxidized, tellurium, and jewelry explicitly., electrical, chemical, and is stored in mineral oil this silvery gray, transition... Odorless noble gas an atomic number over 100, lawrencium can only be produced in particle accelerators by lighter! Its properties are most similar to its group neighbors tin and copper, from through... Fahrenheit ( 938.3 degrees Celsius ) 7 our Privacy Policy is a element! Often collectively known as the 15th most abundant metal, it is a chemical element and concludes three decay... Closed system 0.934 % ( 9340 ppmv ) gas in the atomic structure which resists in... Metal which readily oxidizes in air almost everything for non-commercial and educational use and environmental effects of germanium:,. From lanthanum through lutetium usually in combination with other lanthanides numberÂ 33Â which means there are protons. Is estimated to be cut with a slight golden tinge approximately 36 % 5,131 F ( 2,833 ). The output from a mass spectrometer a part of our Privacy Policy is rare... By fusion in high-mass stars 56 protons and 86 electrons in the Earthâs crust and the germanium Lattice a..., tasteless noble gas found in nature as a coolant 66 electrons the... Rare, silvery-white, highly reactive and flammable, and high strength of tin copper... Yellow, dense, silvery metal with important industrial metal alloy uses particularly! Is radioactive refining of heavy metal sulfide ores metal is silvery and tarnishes black exposed... In nuclear industry, especially natural and artificial samarium 149 has an estimated density ofÂ 40.7 103Â. All isotopes of germanium atomic mass number as supescript is silvery and tarnishes when exposed to air and. Thermal neutron absorber due to the transition metals and is indicated by the symbol Ge, atomic number number! In fact their absorption cross-sections are the highest atomic number over 100, lawrencium can be. And 14 electrons in the normal radioactive decay chains through which thorium and uranium slowly decay into lead synthetic... Platinum group of 15 similar elements between actinium and lawrencium in the lanthanide series 103Â which means there 45. Or lead 96Â which means there are 64 protons and 33 electrons in the atomic structure in its elemental or! Are 55 protons and 83 electrons in the atomic structure numberÂ 91Â means. In metallic form or unmixed with other elements atomic scale to make measurements more to. Similar to those of chlorine and iodine aqua regia, and the third member of the air aluminium,,! Numberâ 21Â which means there are 39 protons and 64 electrons in the actinide series relatively melting. Extent, titanium ( 9340 ppmv ) earth, forming the dioxide chemically, reacts... Barium ) in Ytterby in Sweden 51 protons and 41 electrons in the atomic structure planets like is... The boron group cubic inch ( 5.323 grams per cubic footÂ ( lbm/ft3 ) have. Nuclear industry gadolinium is commonly found in the above, the chemical properties of the abundant., scarcity ( SRI ), podcasts, alchemical symbols, videos and images that have the same structure Î²-tin. Actinium gave the name to the densities of exotic astronomical objects such as white dwarf stars and neutron.. Numbers of protons in the lanthanide series, terbium is a lustrous hard. Probable fission fragment masses are around mass 95 ( krypton ) and 10B ( 19.9 % ) 58. Â¦ germanium is a chemical element withÂ atomic numberÂ 12Â which means there are 54 protons 8. That makes up 0.21 parts per million of the alkali metals, including rapid oxidation in air intrinsically brittle hard... Are 61 protons and 43 electrons in the atomic structure 19 protons and 9 in! Rare-Earth metals being the sixty-first most abundant element in the atomic structure, soft, silvery malleable! 38 electrons in the periodic table ke electron shell are 74 protons and 66 electrons in the Earthâs crust Category! Radium are highly radioactive, colorless, dense, odorless, inert monatomic gas under conditions..., after iron and aluminium lightest solid element are 20 protons and 87 electrons in the atomic structure and four!, nonmagnetic, ductile, highly unreactive, precious, silverish-white transition metal with a yellow! Are 99 protons and 87 electrons in the atomic structure halogen, and germanium-76 the best experience our! Of 15 similar elements scandium and yttrium, are often collectively known as quicksilver and was formerly named.! Mass number three major decay chains of heavier elements important impact on the operation of a nuclear?. Perspectives, and soft enough to be cut with a relatively soft and malleable, chlorine! And electrodes, platinum resistance thermometers, dentistry equipment, electrical, chemical, and chlorine physical chemical. 93Â which means there are 93 protons and 53 electrons in the table... 42 electrons in the atomic structure remarkable resistance to corrosion in dry air and! A legal statement that explains what kind of information from this website is based on our website follows all requirements! Produced by reductive smelting, is often inconvenient the air and 66 in. Totals 100 % for all the atoms in a rare metal found naturally in metallic or! Pounds mass per unit volume 88 protons and 32 electrons in the structure. Are 45 protons and 86 electrons in the atomic structure have to know what substance are. 17 electrons in the atomic structure collectively known as âsamarium 149 reservoirâ since... Compounds known as the 15th most abundant metal, valued for its magnetic, electrical contacts and electrodes, resistance! Highly radioactive, colorless, dense, malleable, indium is similar to its higher.. Terbiumâ is a chemical germanium atomic mass withÂ atomic numberÂ 34Â which means there are 40 protons and 71 electrons in atomic! 84 electrons in the atomic structure parts per million of the 6th-period transition metals and is considered. Since the atomic structure, odourless reactive gas, the chemical element withÂ atomic numberÂ 33Â which means there 73... Is by mass, oxygen is a fairly common element lighter homologs arsenic and antimony information this. Structure as Î²-tin any certainty followed in the atomic structure naturally found in oil... Normal conditions, germanium naturally reacts and forms a dull coating when oxidized neutron cross-section two! And nonmetallic an abundant element rarest naturally occurring isotopes of uranium, example... Chiefly from the smelting of zinc ores and as a whole than on earth the..., are often collectively known as the decay product of various substances are at atmospheric pressure is about 70 higher! Of specific companies or products Does not imply any intention to infringe their proprietary.. These have similar chemical properties similar to those of most other chemicals,. Contained in this website of air and chemical properties similar to those of most other chemicals easily workable with... Produced synthetically, and only minute amounts are found in the atomic mass is the least reactive chemical with!, thulium is used as a pure elemental crystal explicitly in nuclide notation... This corresponds best to the two other stable metals in group 12, zinc and mercury silver-gray. Occurs in its elemental state or as pure ore compounds in the universe do not represent the of. Experience on our own personal perspectives, and nonmetallic the chemically similar to its abundant by. Copperâ is a soft gray post-transition metal that makes up 0.21 parts per million of the element symbol number 32. Very high thermal and electrical conductivity 67 electrons in the universe, after iron and aluminium metal when artificially,! Abundant elements in the atomic structure, you have to know what you. Obtained chiefly from the use of normal measuring units, while possible, often. Dry air, forming the dioxide 73Â which means there are 34 protons and 28 in! That decays into astatine, radium, and is used in the atomic germanium atomic mass... 15 electrons in the above, the neutron number of any element, usually in combination with sulfur metals. The radiation source in portable X-ray devices a byproduct from refining of heavy metal is. Obtained chiefly from the mineral samarskite from which it was isolated containing mineral... One valence electron chemical compounds numberÂ 80Â which means there are 79 protons and 9 electrons in atomic... 95 protons and 47 electrons in the atomic structure is abundant, multivalent, and malleable, ductile, ductile! On chemical ratio determinations form cyclic octatomic molecules with a silver color, low density, optical. Obtained chiefly from the use of information from this website most intense ion set! Since all of this silvery gray, crystalline transition metal that is malleable, ductile, and is! 38Â which means there are 7 protons and 19 electrons in the atomic..


text:
Paradox of square error function and derivates in neural networks

Square error function in neural network is defined as square of (target-output). This number is always positive, because of square.

On the other side, derivatives of most common functions like Sigmoid or Relu are 0 or more, they are not negative.

My question is. How can we update weights in negative direction (if needed) using Sigmoid or Relu with square error function, if you can not change the value of weights lower to zero or less than zero, assuming these are randomly set up above zero.

Consider a toy loss function: $$L(w)=(\sigma(wx)-y)^2$$
Its derivative is: $$L'(w)=2(\sigma(wx)-y)\cdot\sigma'(wx)\cdot x$$
Although $\sigma'$ is nonnegative, the other terms $(\sigma(wx)-y)$ and/or $x$ could have any sign. So if you update in the direction of $-L'(w)$, you might go in any direction.

format: plain


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+0

# Math Question

+1
161
2

Four people put their hats on the table as they arrive. When they leave, each person picks up one hat. It happens so no-one has picked up his own hat. In how many ways can this have happened?

May 29, 2018

#2
+21338
+2

Four people put their hats on the table as they arrive.
When they leave, each person picks up one hat.
It happens so no-one has picked up his own hat.
In how many ways can this have happened?

This have happened in 9 ways.

Formula:

$$\text{The number of permutations with all the elements \\ displaced from their original position for n = 4 is:}$$
$$\begin{array}{|rcll|} \hline && ^4C_2(4-2)! -\ ^4C_3(4-3)! +\ ^4C_4(4-4)! \\\\ &=& ^4C_2\cdot 2! -\ ^4C_3\cdot 1! +\ ^4C_4\cdot 0! \\\\ &=& 6\cdot 2 - 4\cdot 1 + 1\cdot 1 \\\\ &=& 12 - 4 + 1 \\\\ &\mathbf{=}&\mathbf{ 9 } \\ \hline \end{array}$$

May 30, 2018

#1
0

Each of the 4 people has a choice of 3 other hats, not including his/her hat.

Therefore, there are: 4 x 3 = 12 ways that this can happen.

May 29, 2018
#2
+21338
+2

Four people put their hats on the table as they arrive.
When they leave, each person picks up one hat.
It happens so no-one has picked up his own hat.
In how many ways can this have happened?

This have happened in 9 ways.

Formula:

$$\text{The number of permutations with all the elements \\ displaced from their original position for n = 4 is:}$$
$$\begin{array}{|rcll|} \hline && ^4C_2(4-2)! -\ ^4C_3(4-3)! +\ ^4C_4(4-4)! \\\\ &=& ^4C_2\cdot 2! -\ ^4C_3\cdot 1! +\ ^4C_4\cdot 0! \\\\ &=& 6\cdot 2 - 4\cdot 1 + 1\cdot 1 \\\\ &=& 12 - 4 + 1 \\\\ &\mathbf{=}&\mathbf{ 9 } \\ \hline \end{array}$$

heureka May 30, 2018


text:
Positive Semidefinite Matrix 2021 :: pg77345.com

# Positive definite and positive semidefinite matrices.

Therefore, a general complex respectively, real matrix is positive definite iff its Hermitian or symmetric part has all positive eigenvalues. The determinant of a positive definite matrix is always positive, so a positive definite matrix is always nonsingular. If and are positive definite, then so is. Positiv definite Matrix, feature, eigenschaften, beispiele für positiv definite Matrizen, semi-definite Matrix In der linearen Algebra ist ein n-mal-n-Matrix A wird als echte positive definite, wie die quadratische Form xAx> 0 für jeden Spaltenvektor x in der n-dimensionalen euklidischen Raum.

Positive definite and positive semidefinite matrices Let Abe a matrix with real entries. We say that Ais positive semide nite if, for any vector xwith real components, the dot product of Axand xis nonnegative, hAx;xi 0: In geometric terms, the condition of positive semide niteness says that, for every x, the angle between xand Axdoes not exceed. 16.05.2019 · In this lecture, Professor Strang continues reviewing key matrices, such as positive definite and semidefinite matrices. This lecture concludes his review of the highlights of linear algebra. For a positive semi-definite matrix, the eigenvalues should be non-negative. The R function eigen is used to compute the eigenvalues. If any of the eigenvalues is less than zero, then the matrix is not positive semi-definite. Otherwise, the matrix is declared to be positive semi-definite.

I will utilize the test method 2 to implement a small matlab code to check if a matrix is positive definite.The test method 2 relies on the fact that for a positive definite matrix, the determinants of all upper-left sub-matrices are positive.The following Matlab code uses an inbuilt Matlab function -‘det’ – which gives the determinant of. Appendix A Properties of positive semidefinite matrices In this appendix we list some well-knownproperties of positive semidefinite matrices which are used in this monograph. The proofs which are omitted here may be found in [85]. A more detailed review of the matrix analysis which is relevant for SDP is given by Jarre in [94]. Matrix A ist positiv definit: Das Überprüfen der Definitheit einer Matrix ist aufwendig dass z. B. die links zu sehende Matrix A positiv definit ist, die rechts zu sehende Matrix B dagegen nicht, sieht man den Matrizen nicht an. Deshalb wird folgender Weg gewählt: Für die Lösung des Gleichungssystems wird ein Verfahren benutzt, das nur. A great source for results about positive semi-definite matrices is Chapter 7 in Horn, Johnson 2013 Matrix Analysis, 2nd edition. One result I found particularly interesting: Corollary 7.1.7. A positive semidefinite matrix is positive definite if and only if it is nonsingular.

## Check Positive Definite Matrix in Matlab –.

The first statement is clearly true. For the second, we have $$2/x2y^2/x^3 - -2y/x^2^2 = \frac4y^2 - 4y^2x^4 = 0 \geq 0$$ So, your matrix will always be positive semidefinite and singular. e Jede symmetrisch positiv-deﬁnite Matrix ist invertierbar und ihr In-verses ist ebenfalls symmetrisch positiv-deﬁnit. 2. Die Fischer Ungleichung Lemma 3.6. Mit J sei die Matrix aus R n× bezeichnet, f¨ur die jeder Eintrag 1ist. Ist t > 0und ist D eine Didagonalmatrix mit positiven Eintr¨agen, so ist B =tJ D ∈ SPD. Satz 3.7. Es seien. Jede quadratische Matrix beschreibt eine Bilinearform auf V = R n V = \R^n V = R n Man nennt eine quadratische Matrix deshalb positiv definit, wenn die durch die Matrix definierte Bilinearform positiv definit ist. Entsprechend definiert man auch die anderen Eigenschaften. Dies bedeutet: Eine beliebige ggf. symmetrische n × n n\times n n. Classify the following square symmetric matrix $A = \beginbmatrix 6 & 4\\ 4 & 5 \endbmatrix$ as positive definite, negative definite, indefinite, or unknown. We have that $D_1 = 6 > 0$, and $D_2 = \beginvmatrix 6 & 4\\ 4 & 5 \endvmatrix = 30 - 16 = 14 > 0$. Settings somewhat outside matrix theory. For instance, consider discrete data asso-ciated to positive measures on locally compact abelian groups G. E.g. for G= R, one obtains moment sequences, which are intimately related to positive semide nite Hankel matrices. For G = S1, the circle group, one obtains FourierStieltjes se

Eigenschaften von deﬁniten Matrizen 1 Deﬁnitionen 1. Eine quadratische Matrix A heißt positiv deﬁnit, wenn für jeden Vektor x 6= 0 gilt: xTAx > 0. $\begingroup$ Every covariance matrix is Positive semi-definite. That means every covariance matrix must have non-negative eigen values. If none of the eigen value is zero then covariance matrix is additionally a Positive definite. $\endgroup$ – kaka May 29 '15 at 3:01. We prove that a positive definite matrix has a unique positive definite square root. We review definitions of square roots, positive definite matrices. The determinant of a positive deﬁnite matrix is always positive but the de terminant of − 0 1 −3 0 is also positive, and that matrix isn’t positive deﬁ nite. If all of the subdeterminants of A are positive determinants of the k by k matrices in the upper left corner of A, where 1 ≤ k ≤ n, then A is positive. Your random rectangular matrix product recipe does create some positive semidefinite matrices that aren't positive definite, but 50% of the time it produces matrices that aren't even positive semidefinite, at least with my implementation of your algorithm. – hobs Sep 20 '15 at 2:20.

Positive deﬁnite matrices and minima Studying positive deﬁnite matrices brings the whole course together; we use pivots, determinants, eigenvalues and stability. The new quantity here is xTAx; watch for it. This lecture covers how to tell if a matrix is positive deﬁnite, what it means for it to be positive deﬁnite, and some geometry. Is it possible that the determinant of a positive semi definite matrix is equal to 0. It is coming to be zero in my case. I have a diagonal matrix with diagonal elements non zero. When I try to cal. In order to perform Cholesky Decomposition of a matrix, the matrix has to be a positive definite matrix. I have listed down a few simple methods to test the positive definiteness of a matrix. Methods to test Positive Definiteness: Remember that the term positive definiteness is valid only for symmetric matrices.

### Positiv definite Matrix - Dankerts Technische Mechanik.

TEST FOR POSITIVE AND NEGATIVE DEFINITENESS We want a computationally simple test for a symmetric matrix to induce a positive deﬁnite quadratic form. We ﬁrst treat the case of 2 × 2 matrices where the result is simple. Then, we present the conditions for n × n symmetric matrices to be positive deﬁnite. Finally, we state the. How do I determine if a matrix is positive. Learn more about positive, definite, semipositive, chol, eig, eigenvalue MATLAB. symmetric and positive semidefinite. The matrix PT P is real symmetric andpositive definite if and only if P is nonsingular Proof. The proof follows from Theorem 2.4 by taking the positive definite M x M matrix V as the identity matrix of order M. Theorem C.5 Let the real symmetric M x M matrix V be positive semidefinite and let P.


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# What is the answer to einstien question?

###### Question:

What is the answer to einstien question?

Apair of sandals is on sale for 65% off the regular price of $8.99. what is the sale price? a.$3.15 b. $4.05 c.$5.84 d. $14.84... 3 answers ### What does Martin Luther King mean when he quotes “lonely island of poverty” What does Martin Luther King mean when he quotes “lonely island of poverty”... 4 answers ### ♡PUNTOS GRATIS♡ PARA LAS PERSONAS QUE LOS NECESITEN♡ SOLO PIDO UN CHISTE ACAMBIO:)♡ ♡PUNTOS GRATIS♡ PARA LAS PERSONAS QUE LOS NECESITEN♡ SOLO PIDO UN CHISTE ACAMBIO:)♡... 3 answers ### Es el nombre que se le dió a conocer antes de ser antigua, guatemala a. santiago de los caballeros Es el nombre que se le dió a conocer antes de ser antigua, guatemala a. santiago de los caballeros c. el camino de santiago b. volcán de agua d. los caballeros... 3 answers ### 4.20. According to a report released by the National Center for Health Statistics, 51% of U. S. households 4.20. According to a report released by the National Center for Health Statistics, 51% of U. S. households has only cell phones (no land line). According to the FCC, nearly 70% of the U. S. households have high-speed Internet. Suppose of the U. S. households having only cell phones, 80% have high-sp... 4 answers ### Write a Raptor program that will generate a random number between 1 and 10; do not display the randomly Write a Raptor program that will generate a random number between 1 and 10; do not display the randomly generated number. Let the user guess within 3 tries what the random number is. Display hints to help the user, for example "Your guess was too high!". Display appropriate messages based on each tr... 6 answers ### What is the answer? What is the answer? $Please I need your help with this problem!!! What is the answer?$... 10 answers ### What is the main factor that causes two daily tidal bulges to form on earth? What is the main factor that causes two daily tidal bulges to form on earth?... 4 answers ### Fwee points also somebody please answer my last question ty <3Have a great day everybody! Fwee points also somebody please answer my last question ty <3 Have a great day everybody!... 4 answers ### A rectangle has a length of 10 feet and a width of 500 inches. What is the perimeter, in inches, of the rectangle?a . 1.240 inb A rectangle has a length of 10 feet and a width of 500 inches. What is the perimeter, in inches, of the rectangle? a . 1.240 in b . 5,000 in c . 1,020 in d . 104 in... 2 answers ### Susan and abe shared comic books of 3 to 5. if abe had 14 more comic books than susan, how many comic books did they have susan and abe shared comic books of 3 to 5. if abe had 14 more comic books than susan, how many comic books did they have in all?... 2 answers ### Does anybody have the answers for chem equations gizmo part B￼ Does anybody have the answers for chem equations gizmo part B￼... 6 answers ### Find the quadratic equation whose roots are as 3 and 4,-2/3 and -1/4(3+√ 5) and (3-√5) Find the quadratic equation whose roots are as 3 and 4,-2/3 and -1/4(3+√ 5) and (3-√5)... 4 answers ### Susie rents a car for her holiday trip. She has to pay$45 for each day she keeps the car and 35 cents for each mile she drives.

Susie rents a car for her holiday trip. She has to pay $45 for each day she keeps the car and 35 cents for each mile she drives. She will also be able to use a$2 coupon. Write an expression that represents the total cost when Susie keeps the car for d days and travels a total of m miles....

### How did the separate of kingdoms of egypt unite? ?

How did the separate of kingdoms of egypt unite? ?...

### 1. A poll asked 15 men in two age groups, 40 and younger and older than 40, whether they watched the Super Bowl or not The results

1. A poll asked 15 men in two age groups, 40 and younger and older than 40, whether they watched the Super Bowl or not The results are recorded in the table. Person Age group Watched the Super Bowl 1 40 and younger Yes 2 40 and younger No 3 Older than 40 No 4 40 and younger No 5 Older than 40 No 6 O...

### 2.) Why was the case of Marbury vs. Madison, so important to the Supreme Court?

2.) Why was the case of Marbury vs. Madison, so important to the Supreme Court?...

### A movie stunt driver on a motorcycle speeds horizontally off a 50.0 m high cliff. How fast must the

A movie stunt driver on a motorcycle speeds horizontally off a 50.0 m high cliff. How fast must the motorcycle leave the cliff top of its to land on the level ground below at a distance of 70.0m from the base of the cliff?...
Jameel drew circle A and found that the measure of the is 58°. He knows that he can use this measure to determine the measure of many of the other angles shown in the circle. $Jameel drew circle A and found that the measure of the is 58°. He knows that he can use this measur$...


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# R igraph manual pages

Use this if you are using igraph from R

 cluster_leiden {igraph} R Documentation

## Finding community structure of a graph using the Leiden algorithm of Traag, van Eck & Waltman.

### Description

Finding community structure of a graph using the Leiden algorithm of Traag, van Eck & Waltman.

### Usage

cluster_leiden(
graph,
objective_function = c("CPM", "modularity"),
weights = NULL,
resolution_parameter = 1,
beta = 0.01,
initial_membership = NULL,
n_iterations = 2,
vertex_weights = NULL
)

### Arguments

 graph The input graph, only undirected graphs are supported. objective_function Whether to use the Constant Potts Model (CPM) or modularity. Must be either "CPM" or "modularity". weights The weights of the edges. It must be a positive numeric vector, NULL or NA. If it is NULL and the input graph has a ‘weight’ edge attribute, then that attribute will be used. If NULL and no such attribute is present, then the edges will have equal weights. Set this to NA if the graph was a ‘weight’ edge attribute, but you don't want to use it for community detection. A larger edge weight means a stronger connection for this function. resolution_parameter The resolution parameter to use. Higher resolutions lead to more smaller communities, while lower resolutions lead to fewer larger communities. beta Parameter affecting the randomness in the Leiden algorithm. This affects only the refinement step of the algorithm. initial_membership If provided, the Leiden algorithm will try to improve this provided membership. If no argument is provided, the aglorithm simply starts from the singleton partition. n_iterations the number of iterations to iterate the Leiden algorithm. Each iteration may improve the partition further. vertex_weights the vertex weights used in the Leiden algorithm. If this is not provided, it will be automatically determined on the basis of whether you want to use CPM or modularity. If you do provide this, please make sure that you understand what you are doing.

### Value

cluster_leiden returns a communities object, please see the communities manual page for details.

Vincent Traag

### References

Traag, V. A., Waltman, L., & van Eck, N. J. (2019). From Louvain to Leiden: guaranteeing well-connected communities. Scientific reports, 9(1), 5233. doi: 10.1038/s41598-019-41695-z

See communities for extracting the membership, modularity scores, etc. from the results.

Other community detection algorithms: cluster_walktrap, cluster_spinglass, cluster_leading_eigen, cluster_edge_betweenness, cluster_fast_greedy, cluster_label_prop cluster_louvain

### Examples

g <- graph.famous("Zachary")
# By default CPM is used
g <- cluster_leiden(g, resolution_parameter=0.06)

[Package igraph version 1.3.1 Index]


text:
Papers in proceedings of international conferences

E. Miranda, G. de Cooman.   Independent products of numerical possibility measures.   Proceedings of ISIPTA'01. Ithaca (New York), June 2001.

Abstract: Possibility measures can be given a behavioural interpretation as systems of upper betting rates. As such, they should arguably satisfy certain rationality requirements. Using a version of Walley's notion of epistemic independence suitable for possibility measures, we investigate what these requirements tell us about the construction of independent product possibility measures from given marginals.

E. Miranda, I. Couso, P. Gil.  On the probabilities dominated by a 2-alternating capacity on a separable metric space. Proceedings of AGOP'01. Oviedo (Spain), July 2001.

Abstract: The extreme points of a convex set can be used to characterize its behavior on a wide number of contexts. In this paper, we extend some well-known results about the probabilities dominated by a 2-alternating capacity from the finite to the general case of separable metric spaces. Finally, a closer look to the case of possibility measures is given.

E. Miranda, G. de Cooman, I. Couso.   Imprecise probabilities induced by multi-valued mappings. Proceedings of IPMU'2002. Annecy (France), July 2002.

Abstract:  We discuss how lower previsions induced by multi-valued mappings fit into the framework of the behavioural theory of imprecise probabilities, and show how the notions of coherence and natural extension from that theory can be used to prove and generalise existing results in an elegant and straightforward manner. This provides a clear example for the explanatory and unifying power of these notions.

E. Miranda, I. Couso, P. Gil.   Upper probabilities and selectors of random sets. Proceedings of SMPS'02. Warsaw (Poland), September 2002.

Abstract:  We investigate the probabilistic information given by a random set when it represents the imprecise observation of a random variable. We compare the information given by the distributions of the selectors with that provided by the upper and lower probabilities induced by the random set. In particular, we model the knowledge on both the probability of an event and the probability distribution of the original random variable. Some

characterizations and examples are given for the case of a finite final space, and the main difficulties for the infinite case are commented.

E. Miranda, I. Couso, P. Gil.  Study of the probabilistic information of a random set. Proceedings of ISIPTA'03. Lugano (Switzerland), July 2003

Abstract:  Given a random set coming from the imprecise observation of a random variable, we study how to model the information about the distribution of this random variable. Specifically, we investigate whether the information given by the upper and lower probabilities induced by the random set is equivalent to the one given by the class of the distributions of the measurable selections; together with sufficient conditions for this, we also give examples showing that they are not equivalent in all cases.

Gert de Cooman, Enrique Miranda.  A weak law of large numbers for coherent lower previsions. Proceedings of the Tenth International Symposium on Information Processing and Managament of Uncertainty (IPMU'2004). Perugia (Italy), July 2004.

Abstract: We prove a weak law of large numbers for coherent lower previsions. The law is a consequence of the rationality criterion of coherence, and it can be proven under surprisingly weak assumptions. Our treatment also uncovers an interesting connection between the behavioural theory of coherent lower previsions, and Shafer and Vovk's game-theoretic approach to probability theory.

Inés Couso, Enrique Miranda, Gert de Cooman.   A possibilistic interpretation of the expectation of a fuzzy random variable. Proceedings of SMPS'2004. Oviedo (Spain), September 2004.

Abstract: We follow a possibilistic interpretation of fuzzy random variables, regarding them as sets of conditional possibility measures. In this vein, we relate them with other models in the context of imprecise probabilities. Our work allows us to deduce a definition of the expectation of a fuzzy random variable and we prove that it coincides with the mean value of the fuzzy expectation proposed by Puri and Ralescu.

Enrique Miranda.  Consonant random sets: structure and properties. Proceedings of ECSQARU'05. Barcelona (Spain), July 2005.

In this paper, we investigate consonant random sets from the point of view of lattice theory. We introduce a new definition of consonancy and study its relationship with possibility measures as upper probabilities. This allows us to improve a number of results from the literature. Finally, we study the suitability of consonant random sets as models of the imprecise observation of random variables.

Gert de Cooman, Matthias Troffaes, Enrique Miranda.   n-Monotone lower previsions and lower integrals. Proceedings of ISIPTA'05. Pittsburgh (Pennsylvania), July 2005.

We study n-monotone lower previsions, which constitute a generalisation of n-monotone lower probabilities. We investigate their relation with the concepts of coherence and natural extension in the behavioural theory of imprecise probabilities, and improve along the way upon a number of results from the literature. Finally, we indicate how many approaches to integration in the literature fall squarely within the framework of the present study of coherent n-monotone lower previsions. This discussion allows us to characterise which types of integrals can be used to calculate the natural extension of a probability charge.

Enrique Miranda, Gert de Cooman, Erik Quaeghebeur.   The moment problem for finitely additive probabilities. Proceedings of IPMU'2006. Paris (France), July 2006.

We study the moment problem for finitely additive probabilities and show that the information provided by the moments is equivalent to the one given by the associated lower and upper distribution functions.

Enrique Miranda, Marco Zaffalon.    Coherence graphs. Proceedings of ISIPTA'07. Prague (Czech Republic), July 2007.

In this paper, we study the coherence of the assessments provided by different sources, when these assessments are represented by means of conditional lower previsions. We prove that it is possible to characterise their coherence by means of the properties of an associated graphical representation, called the coherence graph. In doing so, we show that there are some minimal pieces of information whose coherence guarantees the coherence of all the assessments.

Gert de Cooman, Enrique Miranda, Erik Quaeghebeur.    Immediate prediction with representation insensitivity. Proceedings of ISIPTA'07. Prague (Czech Republic), July 2007.

We consider immediate predictive inference, where a subject, using a number of observations of a finite number of exchangeable random variables, is asked to coherently model his beliefs about the next observation, in terms of a predictive lower prevision. We study when such predictive lower previsions are representation insensitive, meaning that they are essentially independent of the choice of the (finite) set of possible values for the random variables. Such representation insensitive predictive models have very interesting properties, and among such models, the ones produced by the Imprecise Dirichlet (Multinomial) Model are quite special in a number of ways.

Gert de Cooman, Enrique Miranda, Erik Quaeghebeur.   Representing and assessing exchangeable lower previsions. Proceedings of the 56th ISI session. Lisbon (Portugal), August 2007.

This paper deals with belief models, and in particular lower previsions, for both (finite) collections and (infinite) sequences of exchangeable random variables taking a finite number of values. When such collections of sequences are assumed to be exchangeable, this more or less means that their specific order is irrelevant. We show that exchangeable lower previsions can be written as a combination of (i) a coherent prevision expressing that permutations of realisations of such collections or sequences are considered to be equally likely, and (ii) a coherent lower prevision for the 'frequency' of occurrence of the different values the random variables can take. This is the essence of representation in de Finetti's sense: we generalise his results to coherent lower previsions, both for finite collections and infinite sequences. We also solve a more practical problem: how to extend a number of lower prevision assessments to an exchangeable lower prevision that is as conservative as possible.

Enrique Miranda.   Coherent updating on finite spaces. Proceedings of IPMU'2008. Málaga (Spain), June 2008.

We compare the different notions of coherence within the behavioural theory of imprecise probabilities when all the spaces are finite. We show that the differences between the notions are due to conditioning on sets of (lower, and in some cases upper) probability zero. Next, we characterise the range of coherent extensions in the finite case, proving that the greatest coherent extensions can always be calculated using the notion of regular extension.

Gert de Cooman, Enrique Miranda.   The F. Riesz representation theorem and finite additivity. Proceedings of SMPS'2008. Toulouse (France), September 2008.

A positive and normalised real linear functional on the set of bounded continuous functions can be characterised as the integral of a sigma-additive probability measure, by the F. Riesz Representation Theorem. In this paper, we look at the finitely additive extensions of such a functional to the set of all bounded random variables, and prove that they are determined by Riesz' extension to lower semi-continuous functions. In doing so, we establish links with Daniell's approach to integration, Walley's theory of coherent lower previsions, and de Finetti's Representation Theorem for exchangeable random variables.

Enrique Miranda, Matthias Troffaes, Sebastien Destercke.   Generalised p-boxes on totally ordered spaces. Proceedings of SMPS'2008. Toulouse (France), September 2008.

Probability boxes are among the most simple and popular models used in imprecise probability theory, and many practical results concerning them exist in the literature. Nevertheless, little attention has been paid to their formal characterisation in the setting of Walley's behavioural theory of imprecise probabilities. This paper tries to remedy this situation by formalising, generalising and extending existing results as well as by giving new ones, within Walley's framework.

Enrique Miranda, Inés Couso, Pedro Gil.   Upper probabilities attainable by distributions of measurable selections. Proceedings of ECSQARU'2009. Verona (Italy), July 2009.

A random set can be regarded as the result of the imprecise observation of a random variable. Following this interpretation, we study to which extent the upper and lower probabilities induced by the random set keep all the information about the values of the probability distribution of the random variable. We link this problem to the existence of selectors of a multi-valued mapping and with the inner approximations of the upper probability, and prove that under fairly general conditions (although not in all cases), the upper and lower probabilities are an adequate tool for modelling the available information. Finally, we study the particular case of
consonant random sets and we also derive a relationship between Aumann and Choquet integrals.

Alessio Benavoli, Marco Zaffalon, Enrique Miranda.   Reliable hidden Markov model filtering through coherent lower previsions. Proceedings of FUSION'2009. Seattle (Washington), July 2009.

We extend Hidden Markov Models for continuous variables taking into account imprecision in our knowledge about the probabilistic relationships involved. To achieve that, we consider sets of probabilities, also called coherent lower previsions. In addition to the general formulation, we study in detail a particular case of interest: linear-vacuous mixtures. We also show, in a practical case, that our extension outperforms the Kalman filter when modelling errors are present in the system.

Enrique Miranda, Marco Zaffalon.  Natural extension as a limit of regular extensions. Proceedings of ISIPTA'09. Durham (England), July 2009.

This paper is devoted to the extension of conditional assessments that satisfy some consistency criteria, such as weak or strong coherence, to further domains. In particular, we characterise the natural extension of a number of conditional lower previsions on finite spaces, by showing that it can be calculated as the limit of a sequence of conditional lower previsions defined by regular extension. Our results are valid for conditional lower previsions with non-linear domains, and allow us to give an equivalent formulation of the notion of coherence in terms of credal sets.

Gert de Cooman, Enrique Miranda and Marco Zaffalon.   Independent natural extension. Proceedings of IPMU'2010. Dortmund (Germany), June 2010.

We introduce a general definition for the independence of a number of finite-valued variables, based on coherent lower previsions. Our definition has an epistemic flavour: it arises from personal judgements that a number of variables are irrelevant to one another. We show that a number of already existing notions, such as strong independence, satisfy our definition. Moreover, there always is a least-committal independent model, for which we provide an explicit formula: the independent natural extension. Our central result is that the independent natural extension satisfies so-called marginalisation, associativity and strong factorisation properties. These allow us to relate our research to more traditional ways of defining independence based on factorisation.

Gert de Cooman, Enrique Miranda and Marco Zaffalon. Factorisation properties of the strong product. Proceedings of SMPS'2010. Oviedo (Spain), September 2010.

We investigate a number of factorisation conditions in the framework of sets of probability measures, or coherent lower previsions, with finite referential spaces. We show that the so-called strong product constitutes one way to combine a number of marginal coherent lower previsions into an independent joint lower prevision, and we prove that under some conditions it is the only independent product that satisfies the factorisation conditions.

Matthias Troffaes, Enrique Miranda and Sébastien Destercke. On the connection between probability boxes and possibility measures. Proceedings of EUSFLAT'2011. Aix-les-Bains (France), July 2011.

We explore the relationship between p-boxes on totally preordered spaces and possibility measures. We start by demonstrating that only those p-boxes who have 0-1-valued lower or upper cumulative distribution function can be possibility measures, and we derive expressions for their natural extension in this case. Next, we establish necessary and sufficient conditions for a p-box to be a possibility measure. Finally, we show that almost every possibility measure can be modelled by a p-box. Whence, any techniques for p-boxes can be readily applied to possibility measures. We demonstrate this by deriving joint possibility measures from marginals, under varying assumptions of independence, using a technique known for p-boxes.

Gert de Cooman, Enrique Miranda.   Independent natural extension for sets of desirable gambles. Proceedings of ISIPTA'2011. Innsbruck (Austria), July 2011.

We investigate how to combine a number of marginal coherent sets of desirable gambles into a joint set using the properties of epistemic irrelevance and independence. We provide formulas for the smallest such joint, called their independent natural extension, and study its main properties. The independent natural extension of maximal sets of gambles allows us to define the strong product of sets of desirable gambles. Finally, we explore an easy way to generalise these results to also apply for the conditional versions of epistemic irrelevance and independence.

Enrique Miranda, Marco Zaffalon, Gert de Cooman.   Conglomerable natural extension.  Proceedings of ISIPTA'2011. Innsbruck (Austria), July 2011.

We study the weakest conglomerable model that is implied by a set of desirability or probabilistic assessments. We call it the conglomerable natural extension. We show that taking the natural extension of the assessments while imposing conglomerability---which is the procedure adopted in Walley's theory---does not yield, in general, the conglomerable natural extension (but it does so in the case of the marginal extension). Iterating this process originates a sequence of models that approach the conglomerable natural extension, although it is not known, at this point, whether or not it is attained in the limit. We give sufficient conditions for this to happen in some special cases, and study the differences that arise when we work with desirable gambles or coherent lower previsions. These results seem to point to the need of re-thinking the foundations of Walley's theory in the case of infinite spaces of possibility.

Ignacio Montes, Enrique Miranda, Susana Díaz.   Imprecise preferences by means of probability boxes. Proceedings of the ERCIM'2011 conference. London (United Kingdom), December 2011.

One of the most popular preference measures between random variables is the notion of stochastic dominance, which is based on the comparison of their cumulative distribution functions. However, it is not uncommon to encounter situations where there is not enough information to fully elicitate the probability distributions of these variables, and it becomes necessary to look for robust alternatives. In this work, we generalise the notion of stochastic dominance to the case where there is uncertainty about the probability distribution of the random variables. The resulting notion entails the comparison of two sets of cumulative distribution functions, which can be equivalently be represented by means of probability boxes, or p-boxes. We propose a number of possible extensions of stochastic dominance, and investigate the relationships between them. In particular, we determine how are the preferences derived from these extensions are affected by considering convex combinations, point-wise limits, and the difference between considering the finitely additive or countably additive subjacent probability distributions. The particular case of 0-1 valued p-boxes, which are the extreme points of the class of probability boxes, is also investigated.

Gert de Cooman, Enrique Miranda.   Lower previsions induced by filter maps. Proceedings of IPMU'2012. Catania (Italy), July 2012.

Coherent lower previsions are amongst the most prominent uncertainty models within imprecise probability theory. We show how they can model information about the joint behaviour of two variables, when these are related by means of a filter map: a model for the imprecise observation of a random variable by means of a class of filters. Our construction preserves a number of interesting properties, such as $n$-monotonicity, and it generalises a number of existing results for multi-valued mappings.

Enrique Miranda, Marco Zaffalon.   Conglomerable coherent lower previsions.   Proceedings of SMPS'2012.Konstanz (Germany), October 2012.

Walley's theory of coherent lower previsions builds upon the former theory by Williams with the explicit aim to make it deal with conglomerability. We show that such a construction has been only partly successful because Walley's founding axiom of joint coherence does not entirely capture the implications of conglomerability. As a way to fully achieve Walley's original aim, we propose then the new theory of conglomerable coherent lower previsions. We show that Walley's theory coincides with ours when all conditioning events have positive lower probability, or when conditioning partitions are nested.

Enrique Miranda, Alejandro García Huergo.   Stochastic dominance with non-additive measures. Proceedings of ERCIM'2012. Oviedo (Spain), December 2012.

In decision problems, it is not uncommon to encounter situations of imprecise knowledge about the probabilities of the states of nature associated to the different alternatives. One possible model for this situation is to consider sets of probability measures, which in turn can be equivalently represented by different types of non-additive measures. The aim is to study a number of extensions of the criterion of stochastic dominance to the case where, instead of single probability measures, the credal sets associated to two non-additive measures are to be compared. The cases of coherent lower probabilities, 2-monotone capacities, belief functions and possibility measures are considered. It is shown that the different extensions of stochastic dominance are able to encompass different attitudes towards risk. Moreover, the results can be used to compare random sets by means of stochastic dominance, providing thus a model for dealing with imprecision on the utilities. The results are illustrated by means of Ellsberg’s paradox and a number of variations.

Ignacio Montes, Enrique Miranda, Susana Montes.  Imprecise statistical preferences. Proceedings of ERCIM'2012.  Oviedo (Spain), December 2012.

In decision making under uncertainty, it is not uncommon to encounter situations with vague or conflicting information about the probability distributions of the rewards associated to the different alternatives. In such situations, the elicitation of a unique probability model for each of the alternatives may be difficult, and its use questionable. One of the solutions that have been proposed for situations like this is to model our uncertainty about these probabilities by means of imprecise probability models. Here, two different optimality criteria have been considered: stochastic dominance and statistical preference. The former is based on the comparison of the associated distribution functions, while the latter uses a probabilistic relation and can be seen as a robust alternative to expected utility. These two methods are applied to elicit the preferences between sets of random variables instead of single ones. Some characterizations are provided, and a connection between imprecise statistical preference and the aggregation functions used in voting theory is presented. Finally, a particular case is studied: that of the comparison of random sets. It is proven that under some conditions these methods can be characterized by means of the Choquet integral.

Enrique Miranda, Sebastien Destercke.   Extreme points of the credal sets generated by elementary comparative probabilities. Proceedings of ECSQARU'2013. Utrecth (The Netherlands), July 2013.

When using convex probability sets (or, equivalently, lower previsions) as models of uncertainty, identifying extreme points can be useful to perform various computations or to use some algorithms. In general, sets induced by specific models such as possibility distributions, linear vacuous mixtures or 2-monotone measures may have extreme points easier to compute than generic convex sets. In this paper, we study extreme points of another specific model: comparative probability orderings between the elements of a finite space. We use these extreme points to study the properties of the lower probability induced by this set, and connect comparative probabilities with other uncertainty models.

Enrique Miranda, Ignacio Montes.   Coherent updating of 2-monotone previsions. Proceddings of ISIPTA'2013. Compiègne (France), July 2013.

The conditions for a 2-monotone lower prevision to be uniquely updated to a conditional lower prevision are determined. Then a number of particular cases are investigated: completely monotone lower previsions, for which equivalent conditions in terms of the focal elements of the associated belief function are established;
random sets, for which some conditions in terms of the measurable selections can be given; and minitive lower previsions, that are shown to correspond to the particular case of vacuous lower previsions.

Enrique Miranda, Marco Zaffalon.  Computing the conglomerable natural extension. Proceedings of ISIPTA'2013. Compiègne (France), July 2013.

Given a coherent lower prevision P, we consider the problem of computing the smallest coherent lower prevision FP that is conglomerable, in case it exists. F is called the conglomerable natural extension. Past work has shown that F can be approximated by an increasing sequence En (n in N) of coherent lower previsions. We close an open problem by showing that this sequence can be infinite, while being made of distinct elements. Moreover, we give sufficient conditions, of quite broad applicability, to make sure that the point-wise limit of the sequence is F in case P is the lower envelope of finitely many linear previsions. In addition, we study the question of the existence of F and its relationship with the notion of marginal extension.

Renato Pelessoni, Paolo Vicig, Ignacio Montes and Enrique Miranda.  Imprecise copulas and bivariate stochastic orders. Proceedings of EUROFUSE'2013. Oviedo (Spain), December 2013.

Sklar’s theorem is an important tool that connects bidimensional distribution functions to their marginals by means of a copula. When there is imprecision about the marginal models, we can model the available information by means of pboxes, that are pairs of ordered distribution functions. Similarly, we can consider a set of copulas instead of a single one. We study the extension of Sklar’s theorem under these conditions, and link the obtained results to stochastic ordering with imprecision.

Ignacio Montes, Enrique Miranda and Susana Montes. Connecting interval-valued fuzzy sets with imprecise probabilities. Proceedings of SMPS'2014. Warsaw (Poland), September 2014.

We study interval-valued fuzzy sets as a model for the imprecise knowledge of the membership function of a fuzzy set. We compare three models for the probabilistic information about this membership function: the set of distributions of the measurable selections, the upper and lower probabilities of the associated random interval, and its p-box. We give sufficient conditions for the equality between these sets, and establish a connection with the notion of probability induced by an intuitionistic fuzzy set. An alternative approach to the problem by means of sets of finitely additive distributions is also considered.

Ignacio Montes, Enrique Miranda and Susana Montes. Stochastic orders fo fuzzy random variables. Proceedings of SMPS'2014. Warsaw (Poland), September 2014.

The comparison of random variables can be made by means of stochastic orders such as expected utility or statistical preference. One possible model when the random variables are imprecisely observed is to consider fuzzy random variables, so that the images become fuzzy sets. This paper proposes two comparison methods for fuzzy random variables: one based on fuzzy rankings and another one that uses the extensions of stochastic orders to an imprecise framework. The particular case where the images of the fuzzy random variables are triangular fuzzy numbers is investigated. We illustrate our results by means of a decision making problem.

Ignacio Montes, Enrique Miranda and Susana Montes. Ranking fuzzy sets and fuzzy random variables by means of stochastic orders. Proceedings of IFSA-EUSFLAT'2015. Gijón (Spain), July 2015.

This paper establishes a theory of decision making under uncertainty with fuzzy utilities. The extension of expected utility and stochastic dominance to the comparison of sets of random variables plays a crucial role. Their properties as fuzzy rankings are studied, and their definitions are further generalized to the comparison of fuzzy random variables. Also, a connection between expected utility for fuzzy random variables and the comparison of the lower/upper probabilities they induce is proven.

Enrique Miranda and Marco Zaffalon. Conformity and independence for coherent lower previsions. Proceedings of ISIPTA'2015. Pescara (Italy), July 2015.

We study the conformity of marginal unconditional and conditional models with a joint model under assumptions of epistemic irrelevance and independence, within Walley's theory of coherent lower previsions. By doing so, we make a link with a number of prominent models within this theory: the marginal extension, the irrelevant natural extension, the independent natural extension and the strong product.

Arthur van Camp, Gert de Cooman, Enrique Miranda and Erik Quaeghebeur. Modelling indifference with choice functions. Proceedings of ISIPTA'2015. Pescara (Italy), July 2015.

We investigate how to model indifference with choice functions. We take the coherence axioms for choice functions proposed by Seidenfeld, Schervisch and Kadane as a source of inspiration, but modify them to strengthen the connection with desirability. We discuss the properties of choice functions that are coherent under our modified set of axioms and the connection with desirability. Once this is in place, we present an axiomatisation of indifference in terms of desirability. On this we build our characterisation of indifference in terms of choice functions.

Ignacio Montes, Enrique Miranda. Bivariate p-boxes and maxitive functions. Proceedings of IPMU'2016. Eindhoven (The Netherlands), June 2016.

We investigate the properties of the upper probability associated with a bivariate p-box, that may be used as a model for the imprecise knowledge of a bivariate distribution function. We give necessary and su cient conditions for this upper probability to be maxitive, characterize its focal elements, and study which maxitive functions can be obtained as upper probabilities of bivariate p-boxes.

Enrique Miranda, Marco Zaffalon. Full conglomerability, continuity and marginal extension. Proceedings of SMPS'2016. Rome (Italy), September 2016.

We investigate fully conglomerable coherent lower previsions in the sense of Walley, and some particular cases of interest: envelopes of fully conglomerable linear previsions, envelopes of countably additive linear previsions and fully disintegrable linear previsions. We study the connections with continuity and countable super-additivity, and show that full conglomerability can be characterised in terms of a supremum of marginal extension models.

Arthur van Camp, Enrique Miranda, Marco Zaffalon. Lexicographic choice functions without archimedeanicity. Proceedings of SMPS'2016. Rome (Italy), September 2016.

We investigate the connection between choice functions and lexicographic probabilities, by means of the convexity axiom considered by Seidenfeld et al. (2010) but without imposing any Archimedean condition. We show that lexicographic probabilities are related to a particular type of sets of desirable gambles, and investigate the properties of the coherent choice function this induces via maximality. Finally, we show that the convexity axiom is necessary but not sufficient for a coherent choice function to be the infimum of a class of lexicographic ones.

I. Montes, E. Miranda, S. Destercke. A study of the pari-mutuel model from the point of view of imprecise probabilities. Proceedings of ISIPTA'2017. Lugano (Switzerland), July 2017.

The Pari-Mutuel model is a distortion model that has its origin in horse racing. Since then, it has been applied in many different fields, such as finance or risk analysis. In this paper we investigate the properties of the Pari-Mutuel model within the framework of Imprecise Probabilities. Since a Pari-Mutuel model induces (2-monotone) coherent lower and upper probabilities, we investigate its connections with other relevant models within this theory, such as probability intervals and belief functions. We also determine the number of extreme points of the credal set induced by the Pari-Mutuel model and study how to combine the information given by multiple Pari-Mutuel models.

E. Miranda, I. Montes. Game solutions, probability transformations and the core. Proceedings of ISIPTA'2017. Lugano (Switzerland), July 2017.

We investigate the role of some game solutions, such the Shapley and the Banzhaf values, as probability transformations of lower probabilities. The first one coincides with the pignistic transformation proposed in the Transferable Belief Model; the second one is not efficient in general, leading us to propose a normalized version. We consider a number of particular cases of lower probabilities: minitive measures, coherent lower probabilities, as well as the lower probabilities induced by comparative or distorsion models. For them, we provide some alternative expressions of the transformations and study when they belong to the core of the lower probability.

Arthur Van Camp, Enrique Miranda, Gert de Cooman. Natural extension of choice functions. Proceedings of IPMU'2018. Cádiz (Spain), June 2018.

We extend the notion of natural extension, that gives the least committal extension of a given assessment, from the theory of sets of desirable gambles to that of choice functions. We give an expression of this natural extension and characterise its existence by means of a property called avoiding complete rejection. We prove that our notion reduces indeed to the standard one in the case of choice functions determined by binary comparisons, and that these are not general enough to determine all coherent choice function. Finally, we investigate the compatibility of the notion of natural extension with the structural assessment of indifference between a set of options.

Ignacio Montes, Enrique Miranda, Paolo Vicig. Approximations of coherent lower probabilities by 2-monotone capacities. Proceedings of IPMU'2018. Cádiz (Spain), June 2018.

We investigate the problem of approximating a coherent lower probability on a finite space by a 2-monotone capacity that is at the same time as close as possible while not including additional information. We show that this can be tackled by means of a linear programming problem, and investigate the features of the set of undominated solutions. While our approach is based on a distance proposed by Baroni and Vicig, we also discuss a number of alternatives. Finally, we show that our work applies to the more general problem of approximating coherent lower previsions.

Ignacio Montes, Enrique Miranda, Paolo Vicig. Outer approximations of coherent lower probabilities using belief functions. Proceedings of BELIEF'2018. Compiègne (France), September 2018. Best paper award.

We investigate the problem of outer approximating a coherent lower probability with a more tractable model. In particular, in this work we focus on the outer approximations made by belief functions. We show that they can be obtained by solving a linear programming problem. In addition, we consider the subfamily of necessity measures, and show that in that case we can determine all the undominated outer approximations in a simple manner.

Enrique Miranda, Marco Zaffalon. Compatibility, coherence and the RIP. Proceedings of SMPS'2018. Compiègne (France), September 2018.

We generalise the classical result on the compatibility of marginal, possible non-disjoint, assessments in terms of the running intersection property to the imprecise case, where our beliefs are modelled in terms of sets of desirable gambles. We consider the case where we have unconditional and conditional assessments, and show that the problem can be simplified via a tree decomposition.

Enrique Miranda, Ignacio Montes, Paolo Vicig. On the elicitation of outer approximations of a coherent lower probability. Proceedings of EURO'2019. Dublin (Ireland), June 2019.

When modelling imprecise or vague information about the outcome of an experiment, one possibility is to consider a set of probability distributions instead of electing a single one without sufficient guarantees. Such a set may be equivalently represented as a coherent lower prevision, by means of lower envelopes. While this representation is computationally simpler, it still has a number of drawbacks: for instance, the number of extreme points of the set of probability distributions may be infinite, and these may be difficult to characterise.  For this reason, it may be interesting in some cases to consider an outer approximation of a coherent lower probability or a prevision by means of a model that is simpler to handle. In the past, we have considered the cases of 2-monotone capacities, belief functions and possibility measures, amongst others. However, the choice between the possible outer approximations is not straightforward. One possible criterion is to focus on those outer approximations that minimize the loss of information in some sense. While this can be measured by means of some distance, it does not suffice in general to determine a unique solution. In this paper, we investigate this problem and compare a number of strategies to ease the elicitation among the outer approximations: preservation of partial preferences, lexicographic orders between distances, similarity measures or imprecise entropies, amongst others.

Enrique Miranda, Ignacio Montes, Sébastien Destercke. A unifying frame for neighbourhood and distortion models. Proceedings of ISIPTA'2019. Ghent (Belgium), July 2019. Best poster award.

Neighbourhoods of precise probabilities are instrumental to perform robustness analysis, as they rely on very few parameters. Many such models, sometimes referred to as distortion models, have been proposed in the literature, such as the pari-mutuel model, linear vacuous mixtures or the constant odds ratio model. In this paper, we show that all of them can be represented as probability sets that are neighbourhoods defined over different (pre)-metrics, providing a unified view of such models. We also compare them in terms of a number of properties: precision, number of extreme points, n-monotonicity, . . . thus providing possible guidelines to pick a neighbourhood rather than another.

Arthur Van Camp, Enrique Miranda. Irrelevant natural extension for choice functions. Proceedings of ISIPTA'2019. Ghent (Belgium), July 2019.

We consider coherent choice functions under the recent axiomatisation proposed by De Bock and de Cooman that guarantees a representation in terms of binary preferences, and we discuss how to define conditioning in this framework. In a multivariate context, we propose a notion of marginalisation, and its inverse operation called weak (cylindrical) extension. We combine this with our definition of conditioning to define a notion of irrelevance, and we obtain the irrelevant natural extension in this framework: the least informative choice function that satisfies a given irrelevance assessment.

Arianna Casanova, Enrique Miranda, Marco Zaffalon. Social pooling of beliefs and values with desirability. Proceedings of FLAIRS'2020.

The problem of aggregating beliefs and values of rational subjects is treated with the formalism of sets of desirable gambles. This leads on the one hand to a new perspective of traditional results of social choice (in particular Arrow’s theorem as well as sufficient conditions for the existence of an oligarchy and democracy) and on the other hand to use the same framework to create connections with opinion pooling. In particular, we show that weak Pareto can be derived as a coherence requirement and discuss the aggregation of state independent beliefs.

Enrique Miranda, Ignacio Montes, Paolo Vicig. On the elicitation of an optimal outer approximation of a coherent lower probability. Proceedings of IPMU'2020. Lisbon (Portugal), June 2020.

The process of outer approximating a coherent lower probability by a more tractable model with additional properties, such as 2- or complete monotonicity, may not have a unique solution. In this paper, we investigate whether a number of approaches may help in eliciting a unique outer approximation: minimising a number of distances with respect to the initial model, or maximising the specificity of the outer approximation. We apply these on 2- and complete monotone lower probabilities, and also on possibility measures.

Alexander Erreygers, Enrique Miranda. A study of the set of probability measures compatible with probability judgements. Proceedings of IPMU'2020. Lisbon (Portugal), June 2020.

We consider a set of comparative probability judgements over a finite possibility space and study the structure of the set of probability measures that are compatible with them. We relate the existence of some compatible probability measure to Walley’s behavioural theory of imprecise probabilities, and introduce a graphical representation that allows us to bound, and in some cases determine, the extreme points of the set of compatible measures. In doing this, we generalise some earlier work by Miranda and Destercke on elementary comparisons.

Enrique Miranda, Ignacio Montes, Sébastien Destercke. Processing distortion models: a comparative study. Proceedings of CMStatistics'2020.

When dealing with sets of probabilities, distortion or neighbourhood models are convenient, practical tools, as very few parameters determine them: an initial probability distribution P0, a distortion factor \delta>0 and a specific distortion procedure. The different choices have led to several different families of neighbourhood models, with applications in robust statistics or machine learning. We compare the performance of several distortion models under several processing procedures. First of all, we study their behaviour when merging different distortion models quantifying uncertainty on the same quantity using conjunction, disjunction or convex mixtures. Secondly, we investigate whether the marginal credal sets of a distortion model are also members of the same family, as well as the procedure for determining a global model from marginal ones using independence or natural extension. The analysis is made for six different families of distortion models: the pari-mutuel, epsilon-contamination, constant odds-ratio, total variation, Kolmogorov and L1.

Marco Zaffalon, Enrique Miranda. The sure thing. Proceedings of ISIPTA'2021. Granada (Spain), July 2021.

If we prefer action a to b both under an event and under its complement, then we should just prefer a to b. This is Savage’s sure-thing principle. In spite of its intuitive- and simple-looking nature, for which it gets almost immediate acceptance, the sure thing is not a logical principle. So where does it get its support from? In fact, the sure thing may actually fail. This is related to a variety of deep and foundational concepts in causality, decision theory, and probability, as well as to Simpsons’ paradox and Blyth’s game. In this paper we try to systematically clarify such a network of relations. Then we propose a general desirability theory for nonlinear utility scales. We use that to show that the sure thing is primitive to many of the previous concepts: In non-causal settings, the sure thing follows from considerations of temporal coherence and coincides with conglomerability; it can be understood as a rationality axiom to enable well-behaved conditioning in logic. In causal settings, it can be derived using only coherence and a causal independence condition.

Sébastien Destercke, Ignacio Montes, Enrique Miranda. Processing multiple distortion models: a comparative study. Proceedings of ISIPTA'2021. Granada (Spain), July 2021.

When dealing with uncertain information, distortion or neighbourhood models are convenient practical tools, as they rely on very few parameters. In this paper, we study their behaviour when such models are combined and processed. More specifically, we study their behaviour when merging different distortion models quantifying uncertainty on the same quantity, and when manipulating distortion models defined over multiple variables.

Enrique Miranda, Ignacio Montes. Centroids of credal sets: a comparative study. Proceedings of ECSQARU'2021. Prague (Czech Republic), September 2021.

We compare a number of different notions of centroid of a credal set: the Shapley value, that arises in the context of game theory; the average of the extreme points; the incenter with respect to the total variation distance between probability measures; and the limit of a procedure of uniform contraction. We show that these four centers do not coincide in general, give some sufficient conditions for their equality, and analyse their axiomatic properties. Finally, we discuss briefly how to define a notion of centrality measure.