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Hematocrit	Shear rate relations	Elevated Generally at both sea levels and high altitudes, hematocrit levels rise as children mature. These health-related causes and impacts of elevated hematocrit levels have been reported: Fall in blood plasma levels Dehydration Administering of testosterone supplement therapy In cases of dengue fever, a high hematocrit is a danger sign of an increased risk of dengue shock syndrome. Hemoconcentration can be detected by an escalation of over 20% in hematocrit levels that will come before shock. For early detection of dengue hemorrhagic fever, it is suggested that hematocrit levels be kept under observations at a minimum of every 24 hours; 3–4 hours is suggested in suspected dengue shock syndrome or critical cases of dengue hemorrhagic fever.
Hematocrit	Shear rate relations	Polycythemia vera (PV), a myeloproliferative disorder in which the bone marrow produces excessive numbers of red cells, is associated with elevated hematocrit. Chronic obstructive pulmonary disease (COPD) and other pulmonary conditions associated with hypoxia may elicit an increased production of red blood cells. This increase is mediated by the increased levels of erythropoietin by the kidneys in response to hypoxia.
Hematocrit	Shear rate relations	Professional athletes' hematocrit levels are measured as part of tests for blood doping or erythropoietin (EPO) use; the level of hematocrit in a blood sample is compared with the long-term level for that athlete (to allow for individual variations in hematocrit level), and against an absolute permitted maximum (which is based on maximum expected levels within the population, and the hematocrit level that causes increased risk of blood clots resulting in strokes or heart attacks).
Hematocrit	Shear rate relations	Anabolic androgenic steroid (AAS) use can also increase the amount of RBCs and, therefore, impact the hematocrit, in particular the compounds boldenone and oxymetholone. Capillary leak syndrome also leads to abnormally high hematocrit counts, because of the episodic leakage of plasma out of the circulatory system.
Hematocrit	Shear rate relations	At higher altitudes, there is a lower oxygen supply in the air and thus hematocrit levels may increase over time.Hematocrit levels were also reported to be correlated with social factors that influence subjects. In the 1966–80 Health Examination Survey, there was a small rise in mean hematocrit levels in female and male adolescents that reflected a rise in annual family income. Additionally, a higher education in a parent has been put into account for a rise in mean hematocrit levels of the child.
Hematocrit	Shear rate relations	Lowered Lowered hematocrit levels also pose health impacts. These causes and impacts have been reported: A low hematocrit level is a sign of a low red blood cell count. One way to increase the ability of oxygen transport in red blood cells is through blood transfusion, which is carried out typically when the red blood cell count is low. Prior to the blood transfusion, hematocrit levels are measured to help ensure the transfusion is necessary and safe.
Hematocrit	Shear rate relations	A low hematocrit with a low mean corpuscular volume (MCV) with a high red cell distribution width (RDW) suggests a chronic iron-deficient anemia resulting in abnormal hemoglobin synthesis during erythropoiesis. The MCV and the RDW can be quite helpful in evaluating a lower-than-normal hematocrit, because they can help the clinician determine whether blood loss is chronic or acute, although acute blood loss typically does not manifest as a change in hematocrit, since hematocrit is simply a measure of how much of the blood volume is made up of red blood cells. The MCV is the size of the red cells and the RDW is a relative measure of the variation in size of the red cell population.
Hematocrit	Shear rate relations	Decreased hematocrit levels could indicate life-threatening diseases such as leukemia. When the bone marrow no longer produces normal red blood cells, hematocrit levels deviate from normal as well and thus can possibly be used in detecting acute myeloid leukemia. It can also be related to other conditions, such as malnutrition, water intoxication, anemia, and bleeding. Pregnancy may lead to women having additional fluid in blood. This could potentially lead to a small drop in hematocrit levels.
Kinetic energy penetrator	Kinetic energy penetrator	A kinetic energy penetrator (KEP), also known as long-rod penetrator (LRP), is a type of ammunition designed to penetrate vehicle armour using a flechette-like, high-sectional density projectile. Like a bullet or kinetic energy weapon, this type of ammunition does not contain explosive payloads and uses purely kinetic energy to penetrate the target. Modern KEP munitions are typically of the armour-piercing fin-stabilized discarding sabot (APFSDS) type.
Kinetic energy penetrator	History	Early cannons fired kinetic energy ammunition, initially consisting of heavy balls of worked stone and later of dense metals. From the beginning, combining high muzzle energy with projectile weight and hardness have been the foremost factors in the design of such weapons. Similarly, the foremost purpose of such weapons has generally been to defeat protective shells of armored vehicles or other defensive structures, whether it is stone walls, sailship timbers, or modern tank armour. Kinetic energy ammunition, in its various forms, has consistently been the choice for those weapons due to the highly focused terminal ballistics.
Kinetic energy penetrator	History	The development of the modern KE penetrator combines two aspects of artillery design, high muzzle velocity and concentrated force. High muzzle velocity is achieved by using a projectile with a low mass and large base area in the gun barrel. Firing a small-diameter projectile wrapped in a lightweight outer shell, called a sabot, raises the muzzle velocity. Once the shell clears the barrel, the sabot is no longer needed and falls off in pieces. This leaves the projectile traveling at high velocity with a smaller cross-sectional area and reduced aerodynamic drag during the flight to the target (see external ballistics and terminal ballistics). Germany developed modern sabots under the name "treibspiegel" ("thrust mirror") to give extra altitude to its anti-aircraft guns during the Second World War. Before this, primitive wooden sabots had been used for centuries in the form of a wooden plug attached to or breech loaded before cannonballs in the barrel, placed between the propellant charge and the projectile. The name "sabot" (pronounced SAB-oh in English usage) is the French word for clog (a wooden shoe traditionally worn in some European countries).
Kinetic energy penetrator	History	Concentration of force into a smaller area was initially attained by replacing the single metal (usually steel) shot with a composite shot using two metals, a heavy core (based on tungsten) inside a lighter metal outer shell. These designs were known as armour-piercing composite rigid (APCR) by the British, high-velocity armor-piercing (HVAP) by the US, and hartkern (hard core) by the Germans. On impact, the core had a much more concentrated effect than plain metal shot of the same weight and size. The air resistance and other effects were the same as for the shell of identical size. High-velocity armor-piercing (HVAP) rounds were primarily used by tank destroyers in the US Army and were relatively uncommon as the tungsten core was expensive and prioritized for other applications.
Kinetic energy penetrator	History	Between 1941 and 1943, the British combined the two techniques in the armour-piercing discarding sabot (APDS) round. The sabot replaced the outer metal shell of the APCR. While in the gun, the shot had a large base area to get maximum acceleration from the propelling charge but once outside, the sabot fell away to reveal a heavy shot with a small cross-sectional area. APDS rounds served as the primary kinetic energy weapon of most tanks during the early-Cold War period, though they suffered the primary drawback of inaccuracy. This was resolved with the introduction of the armour-piercing fin-stabilized discarding sabot (APFSDS) round during the 1970s, which added stabilising fins to the penetrator, greatly increasing accuracy.
Kinetic energy penetrator	Design	The principle of the kinetic energy penetrator is that it uses its kinetic energy, which is a function of its mass and velocity, to force its way through armor. If the armor is defeated, the heat and spalling (particle spray) generated by the penetrator going through the armor, and the pressure wave that develops, ideally destroys the target.The modern kinetic energy weapon maximizes the stress (kinetic energy divided by impact area) delivered to the target by: maximizing the mass – that is, using the densest metals practical, which is one of the reasons depleted uranium or tungsten carbide is often used – and muzzle velocity of the projectile, as kinetic energy scales with the mass m and the square of the velocity v of the projectile (mv2/2).
Kinetic energy penetrator	Design	minimizing the width, since if the projectile does not tumble, it will hit the target face first. As most modern projectiles have circular cross-sectional areas, their impact area will scale with the square of the radius r (the impact area being πr2 )The penetrator length plays a large role in determining the ultimate depth of penetration. Generally, a penetrator is incapable of penetrating deeper than its own length, as the sheer stress of impact and perforation ablates it. This has led to the current designs which resemble a long metal arrow.
Kinetic energy penetrator	Design	For monobloc penetrators made of a single material, a perforation formula devised by Wili Odermatt and W. Lanz can calculate the penetration depth of an APFSDS round.In 1982, an analytical investigation drawing from concepts of gas dynamics and experiments on target penetration led to the conclusion on the efficiency of impactors that penetration is deeper using unconventional three-dimensional shapes.The opposite method of KE-penetrators uses chemical energy penetrators. Two types of such shells are in use: high-explosive anti-tank (HEAT) and high-explosive squash head (HESH). They have been widely used against armour in the past and still have a role but are less effective against modern composite armour, such as Chobham as used on main battle tanks today. Main battle tanks usually use KE-penetrators, while HEAT is mainly found in missile systems that are shoulder-launched or vehicle-mounted, and HESH is usually favored for fortification demolition.
Ocean-V RNA motif	Ocean-V RNA motif	The Ocean-V RNA motif is a conserved RNA structure discovered using bioinformatics. Only a few Ocean-V RNA sequences have been detected, all in sequences derived from DNA that was extracted from uncultivated bacteria found in ocean water. As of 2010, no Ocean-V RNA has been detected in any known, cultivated organism.
Cyclophilin	Cyclophilin	Cyclophilins (CYPs) are a family of proteins named after their ability to bind to ciclosporin (cyclosporin A), an immunosuppressant which is usually used to suppress rejection after internal organ transplants. They are found in all domains of life. These proteins have peptidyl prolyl isomerase activity, which catalyzes the isomerization of peptide bonds from trans form to cis form at proline residues and facilitates protein folding.
Cyclophilin	Cyclophilin	Cyclophilin A is a cytosolic and highly abundant protein. The protein belongs to a family of isozymes, including cyclophilins B and C, and natural killer cell cyclophilin-related protein. Major isoforms have been found within single cells, including inside the Endoplasmic reticulum, and some are even secreted.
Cyclophilin	Mammalian cyclophilins	Human genes encoding proteins containing the cyclophilin domain include: PPIA, PPIB, PPIC, PPID, PPIE, PPIF, PPIG, PPIH PPIL1, PPIL2, PPIL3, PPIL4, PPIAL4, PPIL6 PPWD1 Cyclophilin A Cyclophilin A (CYPA) also known as peptidylprolyl isomerase A (PPIA), which is found in the cytosol, has a beta barrel structure with two alpha helices and a beta-sheet. Other cyclophilins have similar structures to cyclophilin A. The cyclosporin-cyclophilin A complex inhibits a calcium/calmodulin-dependent phosphatase, calcineurin, the inhibition of which is thought to suppress organ rejection by halting the production of the pro-inflammatory molecules TNF alpha and interleukin 2.
Cyclophilin	Mammalian cyclophilins	Cyclophilin A is also known to be recruited by the Gag polyprotein during HIV-1 virus infection, and its incorporation into new virus particles is essential for HIV-1 infectivity.
Cyclophilin	Mammalian cyclophilins	Cyclophilin D Cyclophilin D (PPIF, note that literature is confusing, the mitochondrial cyclophilin is encoded by the PPIF gene), which is located in the matrix of mitochondria, is only a modulatory, but may or may not be a structural component of the mitochondrial permeability transition pore. The pore opening raises the permeability of the mitochondrial inner membrane, allows influx of cytosolic molecules into the mitochondrial matrix, increases the matrix volume, and disrupts the mitochondrial outer membrane. As a result, the mitochondria fall into a functional disorder, so the opening of the pore plays an important role in cell death. Cyclophilin D is thought to regulate the opening of the pore because cyclosporin A, which binds to CyP-D, inhibits the pore opening.
Cyclophilin	Mammalian cyclophilins	However, mitochondria obtained from the cysts of Artemia franciscana, do not exhibit the mitochondrial permeability transition pore
Cyclophilin	Clinical significance	Diseases Overexpression of Cyclophilin A has been linked to poor response to inflammatory diseases, the progression or metastasis of cancer, and aging. Cyclophilins as drug targets Cyclophilin inhibitors, such as cyclosporin, are being developed to treat neurodegenerative diseases. Cyclophilin inhibition may also be a therapy for liver diseases.
Dedekind domain	Dedekind domain	In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors. There are at least three other characterizations of Dedekind domains that are sometimes taken as the definition: see below.
Dedekind domain	Dedekind domain	A field is a commutative ring in which there are no nontrivial proper ideals, so that any field is a Dedekind domain, however in a rather vacuous way. Some authors add the requirement that a Dedekind domain not be a field. Many more authors state theorems for Dedekind domains with the implicit proviso that they may require trivial modifications for the case of fields.
Dedekind domain	Dedekind domain	An immediate consequence of the definition is that every principal ideal domain (PID) is a Dedekind domain. In fact a Dedekind domain is a unique factorization domain (UFD) if and only if it is a PID.
Dedekind domain	The prehistory of Dedekind domains	In the 19th century it became a common technique to gain insight into integer solutions of polynomial equations using rings of algebraic numbers of higher degree. For instance, fix a positive integer m . In the attempt to determine which integers are represented by the quadratic form x2+my2 , it is natural to factor the quadratic form into (x+−my)(x−−my) , the factorization taking place in the ring of integers of the quadratic field Q(−m) . Similarly, for a positive integer n the polynomial zn−yn (which is relevant for solving the Fermat equation xn+yn=zn ) can be factored over the ring Z[ζn] , where ζn is a primitive n-th root of unity.
Dedekind domain	The prehistory of Dedekind domains	For a few small values of m and n these rings of algebraic integers are PIDs, and this can be seen as an explanation of the classical successes of Fermat ( m=1,n=4 ) and Euler ( m=2,3,n=3 ). By this time a procedure for determining whether the ring of all algebraic integers of a given quadratic field Q(D) is a PID was well known to the quadratic form theorists. Especially, Gauss had looked at the case of imaginary quadratic fields: he found exactly nine values of D<0 for which the ring of integers is a PID and conjectured that there were no further values. (Gauss' conjecture was proven more than one hundred years later by Kurt Heegner, Alan Baker and Harold Stark.) However, this was understood (only) in the language of equivalence classes of quadratic forms, so that in particular the analogy between quadratic forms and the Fermat equation seems not to have been perceived. In 1847 Gabriel Lamé announced a solution of Fermat's Last Theorem for all n>2 ; that is, that the Fermat equation has no solutions in nonzero integers, but it turned out that his solution hinged on the assumption that the cyclotomic ring Z[ζn] is a UFD. Ernst Kummer had shown three years before that this was not the case already for 23 (the full, finite list of values for which Z[ζn] is a UFD is now known). At the same time, Kummer developed powerful new methods to prove Fermat's Last Theorem at least for a large class of prime exponents n using what we now recognize as the fact that the ring Z[ζn] is a Dedekind domain. In fact Kummer worked not with ideals but with "ideal numbers", and the modern definition of an ideal was given by Dedekind.
Dedekind domain	The prehistory of Dedekind domains	By the 20th century, algebraists and number theorists had come to realize that the condition of being a PID is rather delicate, whereas the condition of being a Dedekind domain is quite robust. For instance the ring of ordinary integers is a PID, but as seen above the ring OK of algebraic integers in a number field K need not be a PID. In fact, although Gauss also conjectured that there are infinitely many primes p such that the ring of integers of Q(p) is a PID, as of 2016 it is not yet known whether there are infinitely many number fields K (of arbitrary degree) such that OK is a PID. On the other hand, the ring of integers in a number field is always a Dedekind domain.
Dedekind domain	The prehistory of Dedekind domains	Another illustration of the delicate/robust dichotomy is the fact that being a Dedekind domain is, among Noetherian domains, a local property: a Noetherian domain R is Dedekind iff for every maximal ideal M of R the localization RM is a Dedekind ring. But a local domain is a Dedekind ring iff it is a PID iff it is a discrete valuation ring (DVR), so the same local characterization cannot hold for PIDs: rather, one may say that the concept of a Dedekind ring is the globalization of that of a DVR.
Dedekind domain	Alternative definitions	For an integral domain R that is not a field, all of the following conditions are equivalent: (DD1) Every nonzero proper ideal factors into primes. (DD2) R is Noetherian, and the localization at each maximal ideal is a discrete valuation ring. (DD3) Every nonzero fractional ideal of R is invertible. (DD4) R is an integrally closed, Noetherian domain with Krull dimension one (that is, every nonzero prime ideal is maximal).
Dedekind domain	Alternative definitions	(DD5) For any two ideals I and J in R , I is contained in J if and only if J divides I as ideals. That is, there exists an ideal H such that I=JH . A commutative ring (not necessarily a domain) with unity satisfying this condition is called a containment-division ring (CDR).Thus a Dedekind domain is a domain that either is a field, or satisfies any one, and hence all five, of (DD1) through (DD5). Which of these conditions one takes as the definition is therefore merely a matter of taste. In practice, it is often easiest to verify (DD4).
Dedekind domain	Alternative definitions	A Krull domain is a higher-dimensional analog of a Dedekind domain: a Dedekind domain that is not a field is a Krull domain of dimension 1. This notion can be used to study the various characterizations of a Dedekind domain. In fact, this is the definition of a Dedekind domain used in Bourbaki's "Commutative algebra". A Dedekind domain can also be characterized in terms of homological algebra: an integral domain is a Dedekind domain if and only if it is a hereditary ring; that is, every submodule of a projective module over it is projective. Similarly, an integral domain is a Dedekind domain if and only if every divisible module over it is injective.
Dedekind domain	Some examples of Dedekind domains	All principal ideal domains and therefore all discrete valuation rings are Dedekind domains.
Dedekind domain	Some examples of Dedekind domains	The ring R=OK of algebraic integers in a number field K is Noetherian, integrally closed, and of dimension one: to see the last property, observe that for any nonzero prime ideal I of R, R/I is a finite set, and recall that a finite integral domain is a field; so by (DD4) R is a Dedekind domain. As above, this includes all the examples considered by Kummer and Dedekind and was the motivating case for the general definition, and these remain among the most studied examples.
Dedekind domain	Some examples of Dedekind domains	The other class of Dedekind rings that is arguably of equal importance comes from geometry: let C be a nonsingular geometrically integral affine algebraic curve over a field k. Then the coordinate ring k[C] of regular functions on C is a Dedekind domain. This is largely clear simply from translating geometric terms into algebra: the coordinate ring of any affine variety is, by definition, a finitely generated k-algebra, hence Noetherian; moreover curve means dimension one and nonsingular implies (and, in dimension one, is equivalent to) normal, which by definition means integrally closed.
Dedekind domain	Some examples of Dedekind domains	Both of these constructions can be viewed as special cases of the following basic result: Theorem: Let R be a Dedekind domain with fraction field K. Let L be a finite degree field extension of K and denote by S the integral closure of R in L. Then S is itself a Dedekind domain.Applying this theorem when R is itself a PID gives us a way of building Dedekind domains out of PIDs. Taking R = Z, this construction says precisely that rings of integers of number fields are Dedekind domains. Taking R = k[t], one obtains the above case of nonsingular affine curves as branched coverings of the affine line.
Dedekind domain	Some examples of Dedekind domains	Zariski and Samuel were sufficiently taken with this construction to ask whether every Dedekind domain arises from it; that is, by starting with a PID and taking the integral closure in a finite degree field extension. A surprisingly simple negative answer was given by L. Claborn.If the situation is as above but the extension L of K is algebraic of infinite degree, then it is still possible for the integral closure S of R in L to be a Dedekind domain, but it is not guaranteed. For example, take again R = Z, K = Q and now take L to be the field Q¯ of all algebraic numbers. The integral closure is nothing else than the ring Z¯ of all algebraic integers. Since the square root of an algebraic integer is again an algebraic integer, it is not possible to factor any nonzero nonunit algebraic integer into a finite product of irreducible elements, which implies that Z¯ is not even Noetherian! In general, the integral closure of a Dedekind domain in an infinite algebraic extension is a Prüfer domain; it turns out that the ring of algebraic integers is slightly more special than this: it is a Bézout domain.
Dedekind domain	Fractional ideals and the class group	Let R be an integral domain with fraction field K. A fractional ideal is a nonzero R-submodule I of K for which there exists a nonzero x in K such that xI⊂R. Given two fractional ideals I and J, one defines their product IJ as the set of all finite sums ∑ninjn,in∈I,jn∈J : the product IJ is again a fractional ideal. The set Frac(R) of all fractional ideals endowed with the above product is a commutative semigroup and in fact a monoid: the identity element is the fractional ideal R. For any fractional ideal I, one may define the fractional ideal I∗=(R:I)={x∈K∣xI⊂R}.
Dedekind domain	Fractional ideals and the class group	One then tautologically has I∗I⊂R . In fact one has equality if and only if I, as an element of the monoid of Frac(R), is invertible. In other words, if I has any inverse, then the inverse must be I∗ A principal fractional ideal is one of the form xR for some nonzero x in K. Note that each principal fractional ideal is invertible, the inverse of xR being simply 1xR . We denote the subgroup of principal fractional ideals by Prin(R).
Dedekind domain	Fractional ideals and the class group	A domain R is a PID if and only if every fractional ideal is principal. In this case, we have Frac(R) = Prin(R) = K×/R× , since two principal fractional ideals xR and yR are equal iff xy−1 is a unit in R.
Dedekind domain	Fractional ideals and the class group	For a general domain R, it is meaningful to take the quotient of the monoid Frac(R) of all fractional ideals by the submonoid Prin(R) of principal fractional ideals. However this quotient itself is generally only a monoid. In fact it is easy to see that the class of a fractional ideal I in Frac(R)/Prin(R) is invertible if and only if I itself is invertible.
Dedekind domain	Fractional ideals and the class group	Now we can appreciate (DD3): in a Dedekind domain (and only in a Dedekind domain) every fractional ideal is invertible. Thus these are precisely the class of domains for which Frac(R)/Prin(R) forms a group, the ideal class group Cl(R) of R. This group is trivial if and only if R is a PID, so can be viewed as quantifying the obstruction to a general Dedekind domain being a PID.
Dedekind domain	Fractional ideals and the class group	We note that for an arbitrary domain one may define the Picard group Pic(R) as the group of invertible fractional ideals Inv(R) modulo the subgroup of principal fractional ideals. For a Dedekind domain this is of course the same as the ideal class group. However, on a more general class of domains, including Noetherian domains and Krull domains, the ideal class group is constructed in a different way, and there is a canonical homomorphism Pic(R) → Cl(R)which is however generally neither injective nor surjective. This is an affine analogue of the distinction between Cartier divisors and Weil divisors on a singular algebraic variety.
Dedekind domain	Fractional ideals and the class group	A remarkable theorem of L. Claborn (Claborn 1966) asserts that for any abelian group G whatsoever, there exists a Dedekind domain R whose ideal class group is isomorphic to G. Later, C.R. Leedham-Green showed that such an R may be constructed as the integral closure of a PID in a quadratic field extension (Leedham-Green 1972). In 1976, M. Rosen showed how to realize any countable abelian group as the class group of a Dedekind domain that is a subring of the rational function field of an elliptic curve, and conjectured that such an "elliptic" construction should be possible for a general abelian group (Rosen 1976). Rosen's conjecture was proven in 2008 by P.L. Clark (Clark 2009).
Dedekind domain	Fractional ideals and the class group	In contrast, one of the basic theorems in algebraic number theory asserts that the class group of the ring of integers of a number field is finite; its cardinality is called the class number and it is an important and rather mysterious invariant, notwithstanding the hard work of many leading mathematicians from Gauss to the present day.
Dedekind domain	Finitely generated modules over a Dedekind domain	In view of the well known and exceedingly useful structure theorem for finitely generated modules over a principal ideal domain (PID), it is natural to ask for a corresponding theory for finitely generated modules over a Dedekind domain.
Dedekind domain	Finitely generated modules over a Dedekind domain	Let us briefly recall the structure theory in the case of a finitely generated module M over a PID R . We define the torsion submodule T to be the set of elements m of M such that rm=0 for some nonzero r in R . Then: (M1) T can be decomposed into a direct sum of cyclic torsion modules, each of the form R/I for some nonzero ideal I of R . By the Chinese Remainder Theorem, each R/I can further be decomposed into a direct sum of submodules of the form R/Pi , where Pi is a power of a prime ideal. This decomposition need not be unique, but any two decompositions T≅R/P1a1⊕⋯⊕R/Prar≅R/Q1b1⊕⋯⊕R/Qsbs differ only in the order of the factors.
Dedekind domain	Finitely generated modules over a Dedekind domain	(M2) The torsion submodule is a direct summand. That is, there exists a complementary submodule P of M such that M=T⊕P (M3PID) P isomorphic to Rn for a uniquely determined non-negative integer n . In particular, P is a finitely generated free module.
Dedekind domain	Finitely generated modules over a Dedekind domain	Now let M be a finitely generated module over an arbitrary Dedekind domain R . Then (M1) and (M2) hold verbatim. However, it follows from (M3PID) that a finitely generated torsionfree module P over a PID is free. In particular, it asserts that all fractional ideals are principal, a statement that is false whenever R is not a PID. In other words, the nontriviality of the class group Cl(R) causes (M3PID) to fail. Remarkably, the additional structure in torsionfree finitely generated modules over an arbitrary Dedekind domain is precisely controlled by the class group, as we now explain. Over an arbitrary Dedekind domain one has (M3DD) P is isomorphic to a direct sum of rank one projective modules: P≅I1⊕⋯⊕Ir . Moreover, for any rank one projective modules I1,…,Ir,J1,…,Js , one has I1⊕⋯⊕Ir≅J1⊕⋯⊕Js if and only if r=s and I1⊗⋯⊗Ir≅J1⊗⋯⊗Js.
Dedekind domain	Finitely generated modules over a Dedekind domain	Rank one projective modules can be identified with fractional ideals, and the last condition can be rephrased as [I1⋯Ir]=[J1⋯Js]∈Cl(R).
Dedekind domain	Finitely generated modules over a Dedekind domain	Thus a finitely generated torsionfree module of rank n>0 can be expressed as Rn−1⊕I , where I is a rank one projective module. The Steinitz class for P over R is the class [I] of I in Cl(R) : it is uniquely determined. A consequence of this is: Theorem: Let R be a Dedekind domain. Then K0(R)≅Z⊕Cl(R) , where K0(R) is the Grothendieck group of the commutative monoid of finitely generated projective R modules.
Dedekind domain	Finitely generated modules over a Dedekind domain	These results were established by Ernst Steinitz in 1912. An additional consequence of this structure, which is not implicit in the preceding theorem, is that if the two projective modules over a Dedekind domain have the same class in the Grothendieck group, then they are in fact abstractly isomorphic.
Dedekind domain	Locally Dedekind rings	There exist integral domains R that are locally but not globally Dedekind: the localization of R at each maximal ideal is a Dedekind ring (equivalently, a DVR) but R itself is not Dedekind. As mentioned above, such a ring cannot be Noetherian. It seems that the first examples of such rings were constructed by N. Nakano in 1953. In the literature such rings are sometimes called "proper almost Dedekind rings".
Explanatory journalism	Explanatory journalism	Explanatory journalism or explanatory reporting is a form of reporting that attempts to present ongoing news stories in a more accessible manner by providing greater context than would be presented in traditional news sources. The term is often associated with the explanatory news website Vox, but explanatory reporting (previously explanatory journalism) has also been a Pulitzer Prize category since 1985. Other examples include The Upshot by The New York Times, Bloomberg Quicktake, The Conversation, and FiveThirtyEight.
Explanatory journalism	Relation to analytic journalism	Journalism professor Michael Schudson says explanatory journalism and analytic journalism are the same, because both attempt to "explain a complicated event or process in a comprehensible narrative" and require "intelligence and a kind of pedagogical flair, linking the capacity to understand a complex situation with a knack for transmitting that understanding to a broad public." Schudson says explanatory journalists "aid democracy."
Space tug	Space tug	A space tug is a type of spacecraft used to transfer spaceborne cargo from one orbit to another orbit with different energy characteristics. An example would be moving a spacecraft from a low Earth orbit (LEO) to a higher-energy orbit like a geostationary transfer orbit, a lunar transfer, or an escape trajectory. The term is often used to refer to reusable, space-based vehicles. Some previously proposed or built space tugs include the NASA 1970s STS proposal or the proposed Russian Parom, and has sometimes been used to refer to expendable upper stages, such as Fregat, or Spaceflight Industries Sherpa.
Space tug	Background	The space tug was first envisioned in the post-World War II era as a support vehicle for a permanent, Earth-orbiting space station. It was used by science fiction writer Murray Leinster as the title of a novel published in 1953 as the sequel to Space Platform, another novel about such a space station.
Space tug	Existing space tugs	Space tugs can be roughly categorised into a few types: Large tugs that dock with satellites in orbit which may be able to perform services like refuelling or repairs or enhancements as well as changing the satellites orbit whether that is to extend life of satellite or to deorbit it. Rocket Kick Stage used to distribute different payloads to different orbits. An example would be Photon Satellite Bus but this might just be considered part of the rocket system rather than a space tug and this article does not really consider these in detail. Smaller tugs that are mainly cubesat deployers with some propulsion to deploy the cubesats to different orbits.
Space tug	Existing space tugs	Large tugs that dock Mission Extension Vehicle In 2011 ViviSat a joint project between U.S. Space and ATK proposed the Mission Extension Vehicle. In 2016 ViviSat was dissolved when U.S. Space declared bankruptcy and ATK merged with Orbital Science Corporation to form Orbital ATK. In 2017 Orbital ATK got the go ahead from the FCC to begin development of the spacecraft with new partner Northrop Grumman who was developing a tug of their own. In June 2018, both companies pooled their resources and merged to form a new company called Northrop Grumman Innovation Systems. On 9 October 2019 the first of these tugs MEV-1 was launched from Baikonur Cosmodrome in Kazakhstan on a Proton-M rocket. In February 2020, MEV-1 successfully docked with Intelsat 901 and returned it to geosynchronous orbit, allowing it to continue operating 4 years past its service life. MEV-1 will continue to maintain this position for a 5-year period, after which it will move the satellite back into a graveyard orbit for retirement. MEV-2 was launched 15 August 2020 with Galaxy 30 on an Ariane 5 to perform a similar maneuver with Intelsat-1002.
Space tug	Existing space tugs	Shijian-21 In December 2021 - January 2022, China's Shijian-21 space debris mitigation satellite has docked with the defunct Beidou-2 G2 navigation satellite to drastically alter its geostationary orbit, demonstrating capabilities only previously exhibited by the United States.
Space tug	Existing space tugs	Smaller tugs and dispensers SHERPA Spaceflight Inc. developed SHERPA, which builds upon the capabilities of the Spaceflight Secondary Payload System (SSPS) by incorporating propulsion and power generation subsystems, which creates a propulsive tug dedicated to maneuvering to an optimal orbit to place secondary and hosted payloads. The maiden flight of two separate unpropelled variants of the dispenser was in December 2018 on a Falcon 9 rocket. This flight deployed 64 small satellites from 17 countries.
Space tug	Existing space tugs	ION Satellite Carrier D-Orbit, an Italian space logistics and transportation company, developed the InOrbit NOW ION Satellite Carrier. The first launch occurred on 3 September 2020 on a Vega rocket, but subsequent launches have all been on SpaceX Falcon 9 Transporter missions. On January 3, 2023 the company launched its seventh and eighth vehicles, Second star to the right, aboard the SpaceX Transporter-6 Mission.
Space tug	Existing space tugs	Long Duration Propulsive ESPA (LDPE) LDPE is based on a Northrop Grumman payload adapter used to help attach the upper stage to the main satellite in addition to hosting a few slots for other smallsats. However, the entire system is powered by the ESPAStar satellite bus, which is in charge of power consumption and distribution as well as propulsion making it a fully operational space tug capable of deploying different payloads at different orbits. ESPAStar has the capability to host 6 smallsat payloads totaling 1,920 kg (4,230 lb). The system is also able to provide 400 meters per second of delta-V via a Hydrazine propulsion module.The first LDPE was launched on 7 December 2021 on an Atlas V rocket as part of the STP-3 mission. The second launch was on 1 November 2022 on a Falcon Heavy rocket as part of the USSF-44 mission. A third mission was on 15 January 2023 on USSF-67 mission.
Space tug	Existing space tugs	Momentus Space Momentus Space develops different space tug versions focusing on large velocity changes over 1 km/s. Two demonstration missions of their Vigoride platform took place on 25 May 2022 and 3 January 2023 with key tests occurring through 2022. Momentus Space became widely known in October 2020 when it reached a SPAC investment deal with Stable Road Acquisition Corp valuing the combined entity at over $1 billion.
Space tug	Existing space tugs	Epic Aerospace Epic Aerospace's Chimera LEO 1 launched on 3 January 2023.
Space tug	Existing space tugs	Launcher Reports surfaced circa June 15, 2021 of Launcher's Orbiter space tug. Launching on its own rocket as well as SpaceX’s Falcon 9 it provides 150 kilograms of payload, either 90 units of CubeSat or else larger satellites using standard smallsat separation systems. With a chemical propulsion system using ethylene and nitrous oxide propellants it is capable of 500 meters per second of delta-v, more with additional propellant tanks. Orbiter SN1 launched on 3 January 2023.
Space tug	Early Concepts - NASA Space Transportation System	A reusable space tug was studied by NASA in the late 60s and early 70s as part of a reusable Space Transportation System (STS). This consisted of a basic propulsion module, to which a crew module or other payload could be attached. Optional legs could be added to land payloads on the surface of the Moon. This, along with all other elements of STS except the Space Shuttle, was never funded after cutbacks to NASA's budget during the 1970s in the wake of the Apollo program.
Space tug	Space Shuttle era	Expendable upper stages The Shuttle program filled the role of high-energy orbital transfer by the development of a solid-fueled single-stage Payload Assist Module and two-stage Inertial Upper Stage.A more powerful liquid hydrogen fueled Centaur-G stage was developed for use on the Shuttle, but was cancelled as too dangerous after the Challenger disaster.
Space tug	Space Shuttle era	Orbital Maneuvering Vehicle NASA studied another space tug design, termed the Orbital Maneuvering Vehicle (OMV), along with its plans for Space Station Freedom. The OMV's role would have been a reusable space vehicle that would retrieve satellites, such as Hubble, and bring them to Freedom for repair or retrieval, or to service uncrewed orbital platforms. In 1984, the Orbital Maneuvering Vehicle (OMV) preliminary design studies were initiated through a competitive award process with systems studies conducted by TRW, Martin Marietta Aerospace, and LTV Corporation.
Space tug	Twenty-first century proposals	Parom The Russian RKK Energia corporation proposed a space tug named Parom in 2005 which could be used to ferry both the proposed Kliper crew vehicle or uncrewed cargo and fuel resupply modules to ISS. Keeping the tug in space would have allowed for a less massive Kliper, enabling launch on a smaller booster than the original Kliper design.
Space tug	Twenty-first century proposals	VASIMR The VASIMR electric plasma rocket could be used to power a high-efficiency space tug, using only 9 tons of Argon propellant to make a round trip to the Moon, delivering 34 tons of cargo from Low Earth Orbit to low lunar orbit. As of 2014, Ad Astra Rocket Company had put forward a concept proposal to utilize the technology to make a space tug.
Space tug	Twenty-first century proposals	ISRO PAM-G Indian Space Research Organisation has built an upper stage called PAM-G (Payload Assist Module for GSLV) capable of pushing payloads directly to MEO or GEO orbits from low Earth orbits. PAM-G is powered by hypergolic liquid motor with restart capability, derived from PSLV's fourth stage. As of 2013, ISRO has realized the structure, control systems, and motors of PAM-G and has conducted hot tests.
Space tug	Twenty-first century proposals	PAM-G would form the fourth stage of GSLV Mk2C launch vehicle, sitting on top of GSLV's cryogenic third stage.
Space tug	Twenty-first century proposals	Jupiter Lockheed Martin made a concept proposal to NASA in 2015 for a design called the Jupiter space tug, to be based on the designs of two earlier Lockheed Martin spacecraft—Mars Atmosphere and Volatile Evolution Mission and the Juno—as well as a robotic arm from MDA derived from technology used on Canadarm, the robotic arm technology previously used on the Space Shuttle. In addition to the Jupiter space tug itself, the Lockheed concept included the use of a new 4.4 m (14 ft)-diameter cargo transport module called Exoliner for carrying cargo to the ISS. Exoliner is based on the earlier (2000s) ESA-developed Automated Transfer Vehicle, and was to be jointly developed with Thales Alenia Space.
Space tug	Twenty-first century proposals	In the event, NASA did not agree to fund the Jupiter development, and Lockheed Martin is not developing the tug with private capital.
Space tug	Twenty-first century proposals	Artemis Transfer Stages One of NASA's Artemis Program's proposed lunar landers is a partially reusable three stage design. One of its main elements is a transfer stage to move the lander from the Lunar Gateway's orbit to a low lunar orbit. Future versions should be able to return to the Gateway for refueling and reuse with another lander. Northrop Grumman has proposed building this transfer stage based on its Cygnus spacecraft. NASA chose to select a different approach in April 2021.
Space tug	Twenty-first century proposals	Moon Cruiser Designed by Airbus, the Moon Cruiser is a conceptual lunar logistics vehicle based on the ATV and ESM that is proposed to be used to support the international Lunar Gateway. If funded, it would make up a part of ESA's contribution to the Lunar Gateway program. As of January 2020, it was in the early design process. Planned to be launched on the Ariane 6—with the capability to also be launched with US heavy launchers: 1:56 —the vehicle is intended to be able to refuel lunar landers and deliver cargo to the Gateway. It will also be used to deliver the European ESPRIT module to the Gateway no earlier than 2025. It has also been proposed to turn the vehicle into a transfer stage for a lunar lander. Concepts for a lander variant of the vehicle exist but have not received funding.
Space tug	Twenty-first century proposals	Skyrora Space Tug British launch vehicle manufacturer Skyrora shared details of their Space Tug in 2021, revealing it to be usable as the third stage of their Skyrora XL rocket. The company shared a video of the Space Tug undergoing a live test in January 2021. As well as being able to move a satellite from one orbit to another the Space Tug can perform a number of in-space operations including space debris removal.
Space tug	Twenty-first century proposals	Exotrail SpaceVan Orbital Transfer Vehicle Exotrail unveils the April 12, 2022 of Orbital Transfer Vehicle, SpaceVan. The debut SpaceVan mission will launch on board a Falcon 9 rideshare mission in October 2023 following a launch service agreement signed between Exotrail and SpaceX. At least three subsequent missions are planned throughout 2024 onboard multiple different launchers. Impulse Space's Mira Demonstration orbital maneuvering and servicing vehicle, Mira is due to launch on SpaceX's Transporter-9 mission in October 2023. Atomos Space In January 2022, Atomos Space announced it had raised $5 million it had been trying to raise since 2020. Atomos plans to launch two of its Quark reusable orbital transfer vehicle in 2023. Firefly Aerospace Firefly Aerospace is developing an OTV called the Space Utility Vehicle that will fly on its Alpha rocket. Space Machine's Optimus In October 2022 Space Machines announced a deal with Arianespace to produce Optimus-1 a 270 kg space tug aiming to launch on SpaceX Falcon 9 in Q2 2023. Exolaunch's Reliant tugs Exolaunch Reliant tugs have standard and pro versions. Testing and flight qualification was planned to begin in 2022 on SpaceX's rideshare missions. Astroscale's Lexi Astroscale is developing Life Extension In-orbit (LEXI™). Orbit Fab Orbit fab is attempting to develop an in-space propellant supply chain aiming to provide 'Gas Stations in Space™'. On 11 January 2022 it was announced they had reached an agreement to refuel Astroscale's LEXI.
Space tug	Twenty-first century proposals	ULA Common Centaur as a SpaceTug The Flexible Lunar Architecture for Exploration (FLARE) is a concept to deliver four crew to the lunar surface for a minimum of seven days and then return them safely to Earth. A key component of FLARE is the modified ULA Common Centaur used as a SpaceTug to deliver an uncrewed human lander to lunar orbit and to assist NASA's Orion capsule returning crew to Earth
Space tug	Other sources	NASA Report, Technical Study for the Use of the Saturn 5, INT-21 and Other Saturn 5 Derivatives to Determine an Optimum Fourth Stage (space tug). Volume 1: Technical Volume, Book 1.
Pitting enamel hypoplasia	Pitting enamel hypoplasia	Enamel hypoplasia can take a variety of forms, but all types are associated with a reduction of enamel formation due to disruption in ameloblast production. One of the most common types, Pitting Enamel Hypoplasia (PEH), ranges from small circular pinpricks to larger irregular depressions. Pits also vary in how they occur on a tooth surface, some forming rows and others more randomly scattered. PEH can be associated with other types of hypoplasia, but it is often the only defect observed. Causes of PEH can range from genetic conditions to environmental factors, and the frequency of occurrence varies substantially between populations and species, likely due to environmental, genetic and health differences. The most striking example of this is in Paranthropus robustus, with half of all primary molars, and a quarter of permanent molars, displaying PEH defects, thought to be caused by a specific genetic condition, amelogenesis imperfecta.
Pitting enamel hypoplasia	Pitting enamel hypoplasia	It is not always clear why PEH forms instead of other hypoplasia types, particularly linear enamel hypoplasia. However, the position on the crown, the tooth type and the cause of the disruption are all likely contributing factors. It has been suggested that because it is relatively rare to have both linear enamel hypoplasia and PEH, these types of defects may be commonly caused by different factors.Each pit is linked to the ceasing of ameloblasts at a particular point in enamel formation. Sometimes, only a couple of ameloblasts stop forming enamel, leading to small PEH defects, with large pits forming when hundreds of these enamel-forming cells stop production. This does not occur in other forms of enamel hypoplasia, such as linear and plane-form, in which all ameloblast activity is affected. Typically with PEH described in archaeological reports, researchers can not specify a cause, with a non-specific stress often concluded. However, in modern clinical studies it is often possible to suggest a cause and these can include the following conditions: hypocalcaemia vitamin D deficiency amelogenesis imperfecta nutritional deficiency maternal diabetes mellitus congenital syphilis premature birth low birth weight hypoparathyroidism neonatal tetany kernicterus
Bushman poison	Bushman poison	Bushman poison can refer to a number of plants or insects used as ingredients by the San people when preparing arrow poisons: Toxicodendron species of the Western Cape province Bushman's poison, Acokanthera spectabilis Bushman's poison, Acokanthera oblongifolia Bushman's poison, Acokanthera oppositifolia Bushman's poison, Acokanthera venenata, of the south and east coasts of South AfricaAlso see genus AcokantheraSucculents: The Gifboom Euphorbias:Euphorbia avesmantana Euphorbia virosaThe Pylgif or Bushman poison:Adenium boehmianumInsects: A beetle genus, Diamphidia
Julian Togelius	Julian Togelius	Julian Togelius is an associate professor at the Department of Computer Science and Engineering at the New York University Tandon School of Engineering.
Julian Togelius	Career	Togelius holds a BA from Lund University, an MSc from the University of Sussex, and a PhD from the University of Essex.He was an associate professor at the Center for Computer Games Research, IT University of Copenhagen before moving to NYU.Togelius is the editor in chief of the IEEE Transactions on Games journal. He is also, with Georgios N. Yannakakis, the co-author of the Artificial Intelligence and Games textbook and the co-organiser of the Artificial Intelligence and Games Summer School series.Togelius co-edited the book Procedural Content Generation Book for games.
Julian Togelius	Research	Togelius was described by Kenneth O. Stanley as one of "the world's most accomplished experts at the intersection of games and AI". His research has appeared in media such as New Scientist, and Le Monde, The Verge, The Economist, and the MIT Technology Review.
Casimir element	Casimir element	In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operator, which is a Casimir element of the three-dimensional rotation group. More generally, Casimir elements can be used to refer to any element of the center of the universal enveloping algebra. The algebra of these elements is known to be isomorphic to a polynomial algebra through the Harish-Chandra isomorphism. The Casimir element is named after Hendrik Casimir, who identified them in his description of rigid body dynamics in 1931.
Casimir element	Definition	The most commonly-used Casimir invariant is the quadratic invariant. It is the simplest to define, and so is given first. However, one may also have Casimir invariants of higher order, which correspond to homogeneous symmetric polynomials of higher order.
Casimir element	Definition	Quadratic Casimir element Suppose that g is an n -dimensional Lie algebra. Let B be a nondegenerate bilinear form on g that is invariant under the adjoint action of g on itself, meaning that ad ad X⁡Z)=0 for all X, Y, Z in g . (The most typical choice of B is the Killing form if g is semisimple.) Let {Xi}i=1n be any basis of g , and {Xi}i=1n be the dual basis of g with respect to B. The Casimir element Ω for B is the element of the universal enveloping algebra U(g) given by the formula Ω=∑i=1nXiXi.
Casimir element	Definition	Although the definition relies on a choice of basis for the Lie algebra, it is easy to show that Ω is independent of this choice. On the other hand, Ω does depend on the bilinear form B. The invariance of B implies that the Casimir element commutes with all elements of the Lie algebra g , and hence lies in the center of the universal enveloping algebra U(g) Quadratic Casimir invariant of a linear representation and of a smooth action Given a representation ρ of g on a vector space V, possibly infinite-dimensional, the Casimir invariant of ρ is defined to be ρ(Ω), the linear operator on V given by the formula ρ(Ω)=∑i=1nρ(Xi)ρ(Xi).
Casimir element	Definition	A specific form of this construction plays an important role in differential geometry and global analysis. Suppose that a connected Lie group G with Lie algebra g acts on a differentiable manifold M. Consider the corresponding representation ρ of G on the space of smooth functions on M. Then elements of g are represented by first order differential operators on M. In this situation, the Casimir invariant of ρ is the G-invariant second order differential operator on M defined by the above formula.
Casimir element	Definition	Specializing further, if it happens that M has a Riemannian metric on which G acts transitively by isometries, and the stabilizer subgroup Gx of a point acts irreducibly on the tangent space of M at x, then the Casimir invariant of ρ is a scalar multiple of the Laplacian operator coming from the metric. More general Casimir invariants may also be defined, commonly occurring in the study of pseudo-differential operators in Fredholm theory. Casimir elements of higher order The article on universal enveloping algebras gives a detailed, precise definition of Casimir operators, and an exposition of some of their properties. All Casimir operators correspond to symmetric homogeneous polynomials in the symmetric algebra of the adjoint representation ad g. :C(m)=κij⋯kXi⊗Xj⊗⋯⊗Xk where m is the order of the symmetric tensor κij⋯k and the Xi form a vector space basis of g. This corresponds to a symmetric homogeneous polynomial c(m)=κij⋯ktitj⋯tk in m indeterminate variables ti in the polynomial algebra K[ti,tj,⋯,tk] over a field K. The reason for the symmetry follows from the PBW theorem and is discussed in much greater detail in the article on universal enveloping algebras. Moreover, a Casimir element must belong to the center of the universal enveloping algebra, i.e. it must obey [C(m),Xi]=0 for all basis elements Xi. In terms of the corresponding symmetric tensor κij⋯k , this condition is equivalent to the tensor being invariant: fijkκjl⋯m+fijlκkj⋯m+⋯+fijmκkl⋯j=0 where fijk are the structure constants of the Lie algebra i.e. [Xi,Xj]=fijkXk
Casimir element	Properties	Uniqueness of the quadratic Casimir element Since for a simple Lie algebra every invariant bilinear form is a multiple of the Killing form, the corresponding Casimir element is uniquely defined up to a constant. For a general semisimple Lie algebra, the space of invariant bilinear forms has one basis vector for each simple component, and hence the same is true for the space of corresponding Casimir operators.
Casimir element	Properties	Relation to the Laplacian on G If G is a Lie group with Lie algebra g , the choice of a nondegenerate invariant bilinear form on g corresponds to a choice of bi-invariant Riemannian metric on G . Then under the identification of the universal enveloping algebra of g with the left invariant differential operators on G , the Casimir element of the bilinear form on g maps to the Laplacian of G (with respect to the corresponding bi-invariant metric).
Casimir element	Properties	Casimir elements and representation theory By Racah's theorem, for a semisimple Lie algebra the dimension of the center of the universal enveloping algebra is equal to its rank. The Casimir operator gives the concept of the Laplacian on a general semisimple Lie group; but there is no unique analogue of the Laplacian, for rank > 1.
Casimir element	Properties	By definition any member of the center of the universal enveloping algebra commutes with all other elements in the algebra. By Schur's Lemma, in any irreducible representation of the Lie algebra, any Casimir element is thus proportional to the identity. The eigenvalues of all Casimir elements can be used to classify the representations of the Lie algebra (and hence, also of its Lie group).Physical mass and spin are examples of these eigenvalues, as are many other quantum numbers found in quantum mechanics. Superficially, topological quantum numbers form an exception to this pattern; although deeper theories hint that these are two facets of the same phenomenon..
Casimir element	Properties	Let L(λ) be the finite dimensional highest weight module of weight λ . Then the quadratic Casimir element Ω acts on L(λ) by the constant ⟨λ,λ+2ρ⟩=⟨λ+ρ,λ+ρ⟩−⟨ρ,ρ⟩, where ρ is the weight defined by half the sum of the positive roots. If L(λ) is nontrivial (i.e. if λ≠0 ), then this constant is nonzero. After all, since λ is dominant, if λ≠0 , then ⟨λ,λ⟩>0 and ⟨λ,ρ⟩≥0 , showing that ⟨λ,λ+2ρ⟩>0 . This observation plays an important role in the proof of Weyl's theorem on complete reducibility. It is also possible to prove the nonvanishing of the eigenvalue in a more abstract way—without using an explicit formula for the eigenvalue—using Cartan's criterion; see Sections 4.3 and 6.2 in the book of Humphreys.

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