* In linear algebra, an endomorphism of a vector space V is a linear operator V → V. An automorphism is an invertible linear operator on V. When the vector space is finite-dimensional, the automorphism group of V is the same as the general linear group, GL ( V ).

When the campaigns in Africa were over, Afonso V found new grounds for battle in the Iberian Peninsula.

When Sultan Murad V began to show signs of paranoia, madness and continuous fainting and vomiting even on the day of his coronation and threw himself into a pool yelling at his guards to protect his life, they were afraid the public would become outraged and revolt to bring the former Sultan back.

When awarded for acts of heroism, the medal is awarded with the " V " Device.

When U and V are two independent normally distributed random variables with expected value 0 and variance 1, then the ratio U / V has the standard Cauchy distribution.

When a system, for example, n moles of a gas of volume V at pressure p and temperature T, is created or brought to its present state from absolute zero, energy must be supplied equal to its internal energy U plus pV, where pV is the work done in pushing against the ambient ( atmospheric ) pressure.

He returned to work for the Florentine government in 1365, undertaking a mission to Pope Urban V. When the papacy returned to Rome from Avignon in 1367, Boccaccio was again sent to Urban, offering congratulations.

When Heracles took the cattle of Geryon, he shot Hera in the right breast with a triple-barbed arrow: the wound was incurable and left her in constant pain, as Dione tells Aphrodite in the Iliad, Book V. Afterwards, Hera sent a gadfly to bite the cattle, irritate them and scatter them.

When James V finally managed to escape from the custody of the regents with the aid of his redoubtable mother in 1528, he once again set about subduing the rebellious Highlands, Western and Northern isles, as his father had had to do.

When the last male member of the Salian dynasty, Emperor Henry V, had died without heirs in 1125, a controversy arose about the succession.

When France rejected Neurath's note, Germany stormed out of the League of Nations and the World Disarmament Conference, and all but announced its intention to unilaterally violate Part V. Consequently, there were several calls in France that autumn for a preventive war to put an end to the Nazi regime while Germany was still more-or-less disarmed.

When the voltage of C1 right-hand plate ( Q2 base voltage ) becomes positive and reaches 0. 6 V, Q2 base-emitter junction begins diverting a part of R2 charging current.

If E < sub > i </ sub > is applied to the inverting input, the circuit is an inverting positive-level detector: When E < sub > i </ sub > is above V < sub > ref </ sub >, V < sub > O </ sub > equals − V < sub > sat </ sub >.

AD 790 and describing events of the 660s: " When Grimuald set out for Beneventum, he entrusted his palace to Lupus " ( Historia Langobardorum, V. xvii ).

When the " nations " of the council pressed their plans for reform, Martin V submitted a counter-scheme and ultimately entered into negotiations for separate concordats, for the most part vague and illusory, with the Holy Roman Empire, England, and France.

When excommunication and interdict failed to have their intended effect, Clement V preached a crusade against the Venetians in May 1309, declaring that Venetians captured abroad might be sold into slavery, like non-Christians, a symptom of how polarized that particular conflict had become.

When the speed of the runner is equal to the initial jet velocity, the torque is zero ( i. e. when u = V < sub > i </ sub >, then T = 0 ).

When there are only two means to compare, the t-test and the ANOVA F-test are equivalent ; the relation between ANOVA and t is given by F = t < sup > 2 </ sup >.

When the volume of an alcoholic drink is shown in centilitres, determining the number of units in a drink is as simple as < var > volume </ var > × < var > percentage </ var > ( converted into a fraction of 1 ).

( When measured in decibels it is logarithmically related to the power ratio: G ( dB )= 10 log ( P < sub > out </ sub > /( P < sub > in </ sub >)).

When fully implemented, CAIR will reduce SO < sub > 2 </ sub > emissions in 28 eastern states and the District of Columbia by over 70 percent and NOx emissions by over 60 percent from 2003 levels.

When clouds are present, the loss rate of SO < sub > 2 </ sub > is faster than can be explained by gas phase chemistry alone.

When certain ionic compounds containing bromine are mixed with potassium permanganate ( KMnO < sub > 4 </ sub >) and an acidic substance, they will form a pale brown cloud of bromine gas.

According to the theorem, it is possible to expand the power ( x + y )< sup > n </ sup > into a sum involving terms of the form ax < sup > b </ sup > y < sup > c </ sup >, where the exponents b and c are nonnegative integers with, and the coefficient a of each term is a specific positive integer depending on n and b. When an exponent is zero, the corresponding power is usually omitted from the term.

When the functions J < sub > n </ sub >( x ) are plotted on the same graph, though, none of the zeros seem to coincide for different values of n except for the zero at x = 0.

When the rate of bone formation exceeds the rate of bone resorption, soft tissue < sup > 44 </ sup > Ca /< sup > 40 </ sup > Ca rises.

When the partial pressure of is reduced, for example when a can of soda is opened, the equilibrium for each of the forms of carbonate ( carbonate, bicarbonate, carbon dioxide, and carbonic acid ) shifts until the concentration of in the solution is equal to the solubility of CO < sub > 2 </ sub > at that temperature and pressure.

When pH is too high, the concentration of hydrogen ions in the blood is too low, so the kidneys excrete bicarbonate ( HCO < sub > 3 </ sub >< sup >−</ sup >).

When applied to the simplest two-electron molecule, H < sub > 2 </ sub >, valence bond theory, even at the simplest Heitler-London approach, gives a much closer approximation to the bond energy, and it provides a much more accurate representation of the behavior of the electrons as chemical bonds are formed and broken.

When this is done, the model radius still decreases with mass, but becomes zero at M < sub > limit </ sub >.

When the critical stress, F < sub > cr </ sub > ( F < sub > cr </ sub > =

When Newton's method is used to find the solution for the maximum likelihood estimate, the middle 24 % order statistics can be used as an initial solution for x < sub > 0 </ sub >.

When s < sup > 2 </ sup >− v < sup > 2 </ sup > is expanded using these expressions the result is

When the trajectory r ( t ) rotates an amount dθ, u < sub > ρ </ sub >, which points in the same direction as r ( t ), also rotates by dθ.

When Yellowstone Caldera last erupted some 650, 000 years ago, it released about 1, 000 km < sup > 3 </ sup > of material ( as measured in dense rock equivalent ( DRE )), covering a substantial part of North America in up to two metres of debris.

* When the metal atoms are hexagonal close packed, ( hcp ), as the octahedral interstices lie directly opposite each other on either side of the layer of metal atoms, filling only one of these with carbon achieves 2: 1 stoichiometry with the CdI < sub > 2 </ sub > structure.

When the DNS resolver gets a PTR ( reverse-lookup ) request, it begins by querying the root servers ( which point to The American Registry For Internet Numbers ' ( ARIN's ) servers for the < tt > 208. in-addr. arpa </ tt > zone ).

When heavy metals or radionuclides form their own pure colloids, the term " Eigencolloid " is used to designate pure phases, e. g., Tc ( OH )< sub > 4 </ sub >, U ( OH )< sub > 4 </ sub >, Am ( OH )< sub > 3 </ sub >.

When Warhol was 13, his father died in an accident .< ref >

Cottard takes advantage of the crisis to make money by selling contraband cigarettes and inferior liquor .</ br > When the epidemic ends, Cottard's moods fluctuate.

However, years of activism, and fighting for the Republican side of the Spanish Civil War have left him disillusioned .</ br > When the plague epidemic is virtually over, Tarrou becomes one of its last victims, but puts up a heroic struggle before dying.

116 .</ ref > When air condenses it becomes visible, as mist and then rain and other forms of precipitation, and as the condensed air cools Anaximenes supposed that it went on to form earth and ultimately stones.