If ( remember this is an assumption ) the minimal polynomial for T decomposes Af where Af are distinct elements of F, then we shall show that the space V is the direct sum of the null spaces of Af.

Let N be a positive integer and let V be the space of all N times continuously differentiable functions F on the real line which satisfy the differential equation Af where Af are some fixed constants.

The photocathode sensitivities S, phosphor efficiencies P, and anode potentials V of the individual stages shall be distinguished by means of subscripts 1, and 2, in the text, where required.

where P is pressure, V is specific or molar volume, and

Alexios attempted to organize a resistance to the new regime from Adrianople and then Mosynopolis, where he was joined by the later usurper Alexios V Doukas Mourtzouphlos in April 1204, after the definitive fall of Constantinople to the crusaders and the establishment of the Latin Empire.

The refugees reached Mosynopolis, the base of the deposed emperor Alexios III Angelos, where they were initially well received, and Alexios V married Eudokia Angelina.

The pirates had raided the Danish coasts during the civil war of Sweyn III, Canute V, and Valdemar, to the point where at the accession of Valdemar one-third of Denmark lay wasted and depopulated.

*( V, x ) where x is a value of unspecified nature ; and

As in so many programming languages, the operation ( V, x ) is often written V ← x ( or some similar notation ), and ( V ) is implied whenever a variable V is used in a context where a value is required.

where R < sub > i </ sub > is the position vector of particle i from the reference point, m < sub > i </ sub > is its mass, and V < sub > i </ sub > is its velocity.

Regulatory Arbitrage was used for the first time in 2005 when it was applied by Scott V. Simpson, a partner at law firm Skadden, Arps, to refer to a new defence tactic in hostile mergers and acquisitions where differing takeover regimes in deals involving multi-jurisdictions are exploited to the advantage of a target company under threat.

Only from Spain did Louis XIV receive any good news where Das Minas and Galway had been forced to retreat from Madrid towards Valencia, allowing Philip V to re-enter his capital on 4 October.

From the 1930s to the 1960s, archaeologists like V. Gordon Childe and later John Hamilton regarded them as castles where local landowners held sway over a subject population.

Such syllables may be abbreviated CV, V, and CVC, where C stands for consonant and V stands for vowel.

Another place where a heterogeneous catalyst is applied is in the contact process ( oxidation of sulfur dioxide on vanadium ( V ) oxide for the production of sulfuric acid ).

A simple electric circuit, where current is represented by the letter i. The relationship between the voltage ( V ), resistance ( R ), and current ( I ) is V = IR ; this is known as Ohm's Law.

( V, T, P, S ), where N / V is the Non-terminal Variable, and Σ / T is the Terminal ) is context-sensitive if all rules in P are of the form

( Indeed, every production of a context free grammar is of the form V → w where V is a single nonterminal symbol, and w is a string of terminals and / or nonterminals ( w can be empty )).

He also offers a conversion table ( see Cohen, 1988, p. 283 ) for eta squared ( η < sup > 2 </ sup >) where 0. 0099 constitutes a small effect, 0. 0588 a medium effect and 0. 1379 a large effect.

Reactions of acids are often generalized in the form HA H < sup >+</ sup > + A < sup >−</ sup >, where HA represents the acid and A < sup >−</ sup > is the conjugate base.

The numerical value of K < sub > a </ sub > is equal to the concentration of the products divided by the concentration of the reactants, where the reactant is the acid ( HA ) and the products are the conjugate base and H < sup >+</ sup >.

Because the range of possible values for K < sub > a </ sub > spans many orders of magnitude, a more manageable constant, pK < sub > a </ sub > is more frequently used, where pK < sub > a </ sub >

These nodes are areas where the action potential is amplified using a high density of sodium ( Na < big >+</ big >) ions and is subsequently passed along the axon.

functions as real combinations of spherical harmonics Y < sub > lm </ sub >( θ, φ ) ( where l and m are quantum numbers ).

Within a shell where n is some integer n < sub > 0 </ sub >, ranges across all ( integer ) values satisfying the relation.

It lacks the NH < sub > 2 </ sub > group because of the cyclization of the side-chain and is known as an imino acid ; it falls under the category of special structured amino acids .</ ref > where R is an organic substituent known as a " side-chain "); often the term " amino acid " is used to refer specifically to these.

½ ( pKa < sub > 1 </ sub > + pKa < sub > R </ sub >), where pKa < sub > R </ sub > is the side-chain pKa.

## < tt > SubBytes </ tt >— a non-linear substitution step where each byte is replaced with another according to a lookup table.

## < tt > ShiftRows </ tt >— a transposition step where each row of the state is shifted cyclically a certain number of steps.

The < tt > MixColumns </ tt > function takes four bytes as input and outputs four bytes, where each input byte affects all four output bytes.

where n < sup > c </ sup > denotes the charge conjugate state, i. e., the antiparticle.

where we use the symbol k to denote the quantum numbers p and σ of the previous section and the sign of the energy, E ( k ), and a < sub > k </ sub > denotes the corresponding annihilation operators.

where E < sub > 0 </ sub > is an infinite negative constant.

The Battle of Qarqar is mentioned in extra-biblical records, and was perhaps at Apamea where Shalmaneser III of Assyria fought a great confederation of princes from Cilicia, Northern Syria, Israel, Ammon, and the tribes of the Syrian desert ( 853 BC ), including Ahab ( A-ha-ab-bu < sup > mat </ sup >) ( Adad -' idri ).

where < big ></ big > is the number of degrees of freedom divided by two, R is the universal gas constant and n is the number of moles in the system ( a constant ).

We could, alternatively, choose an encoding for Turing machines, where an encoding is a function which associates to each Turing Machine M a bitstring < M >.

where p is a program in L < sub > 2 </ sub >.

Following each individual brick should be a layer of adobe mortar, recommended to be at least an inch thick to make certain there is ample strength between the brick ’ s edges and also to provide a relative moisture barrier during the seasons where the arid climate does produce rain.

This theme was greater strengthened by Christie ’ s time spent in the Middle East where she was consistently surrounded by the religious temples and spiritual history of the towns and cities they were excavating in Mallowan ’ s archaeological work.

When more electrons are added to a single atom, the additional electrons tend to more evenly fill in a volume of space around the nucleus so that the resulting collection ( sometimes termed the atom ’ s “ electron cloud ” ) tends toward a generally spherical zone of probability describing where the atom ’ s electrons will be found.

The British refused and only agreed to transfer Hasan Ali Shah ’ s residence to Calcutta, where it would be more difficult for him to launch new attacks against the Persian government.

Alcaeus was a contemporary and a countryman of Sappho and, since both poets composed for the entertainment of Mytilenean friends, they had many opportunities to associate with each other on a quite regular basis, such as at the Kallisteia, an annual festival celebrating the island's federation under Mytilene, held at the ' Messon ' ( referred to as temenos in fr. s 129 and 130 ), where Sappho performed publicly with female choirs.

Asser ’ s account of the raid places the Danish raiders at the Saxon city of Rochester, where they built a temporary fortress in order to besiege the city.

The Kocher enters the town ’ s territory from Oberkochen to the south, crosses the district of Unterkochen, then enters the town centre, where the Aal flows into it.

Cuyp probably first encountered a painting by van Goyen in 1640 when van Goyen was, as Stephen Reiss points out “ at the height of powers .” This is noticeable in the comparison between two of Cuyp ’ s landscape paintings inscribed 1639 where no properly formed style is apparent and the landscape backgrounds he painted two years later for two of his father ’ s group portraits that are distinctly van Goyenesque.

As has been mentioned and as will be explained in depth below, there are pieces where Aelbert provided the landscape background for his father ’ s portraits.

Sakharov then tested a MK-driven " plasma cannon " where a small aluminium ring was vaporized by huge eddy currents into a stable, self-confined toroidal plasmoid and was accelerated to 100 km / s.

On 24 April 1826 over 1, 000 men and women, many armed, gathered at Whinney Hill in Clayton-le-Moors to listen to a speaker from where they marched on Sykes ’ s Mill at Higher Grange Lane, near the site of the modern police station and Magistrate ’ s Courts, and smashed over 60 looms.

Athanasius ' Shrine ( where a portion of his relic s are preserved ) under Coptic Cathedral | St. Mark's Cathedral, Cairo

At the point where the Arno leaves the Apennines, flow measurements can vary between 0. 56 m³ / s and 3, 540 m³ / s.

Thus, many users of a given BBS usually lived in the same area, and activities such as BBS Meets or Get Togethers were common, where users of the board would gather at a local restaurant, the SysOp ’ s home or similar venue and meet face to face.

Devices functioning with GFSK are said to be operating in basic rate ( BR ) mode where an instantaneous data rate of 1 Mbit / s is possible.

When the serving side loses a rally, the serve immediately passes to their opponent ( s ) ( this differs from the old system where sometimes the serve passes to the doubles partner for what is known as a " second serve ").

“ A visit to the ground has only confirmed me ,” Lucas wrote in 1921 ; “ and it was interesting to find that Mr. Apostolides, son of the large local landowner, the hospitality of whose farm at Tekés I enjoyed, was convinced too that the site was by Driskole Krini, for the very sound reason that neither the hills nor the river further east suit Caesar ’ s description .” John D. Morgan in his definitive “ Palae-pharsalus – the Battle and the Town ”, arguing for a site closer still to Krini, where he places Palaepharsalos, writes: “ My reconstruction is similar to Lucas ’ s, and in fact I borrow one of his alternatives for the line of the Pompeian retreat.

The year 1705 proved almost entirely barren for the Duke whose military disappointments were only partly compensated by efforts on the diplomatic front where, at the courts of Düsseldorf, Frankfurt, Vienna, Berlin and Hanover, Marlborough sought to bolster support for the Grand Alliance and extract promises of prompt assistance for the following year ’ s campaign.

However, it seems that the Duke ’ s favoured scheme was to return to the Moselle valley ( where Marshal Marsin had recently taken command of French forces ) and once more attempt an advance into the heart of France.