So assume ƒ is differentiable at 0, as a function of two real variables from Ω to C. This is equivalent to the existence of two complex numbers and ( which are the partial derivatives of ƒ ) such that we have the linear approximation

So Daniele da Volterra, an apprentice of Michelangelo, was commissioned to cover with perizomas ( briefs ) the genitals, leaving unaltered the complex of bodies.

So the logarithm cannot be defined for the whole complex plane, and even then it is multi-valued – any complex logarithm can be changed into an " equivalent " logarithm by adding any integer multiple of 2πi at will.

So far as transport was concerned, he decided that dogs would be one element in a complex strategy that also involved horses and motor sledges, and much man-hauling.

So the syntactic category for an intransitive verb is a complex formula representing the fact that the verb acts as a function word requiring an NP as an input and produces a sentence level structure as an output.

So the realist can claim that, while his world view is more complex, it is more satisfying as an explanation.

So the sphere and torus admit complex structures, but the Möbius strip, Klein bottle and projective plane do not.

So one continuous motion in the complex plane has transformed the positive square root e < sup > 0 </ sup > =

* Spec k, the spectrum of the polynomial ring over a field k, which is also denoted, the affine line: the polynomial ring is known to be a principal ideal domain and the irreducible polynomials are the prime elements of k. If k is algebraically closed, for example the field of complex numbers, a non-constant polynomial is irreducible if and only if it is linear, of the form t − a, for some element a of k. So, the spectrum consists of one closed point for every element a of k and a generic point, corresponding to the zero ideal.

So far, excavations have not revealed whether Netley had a separate infirmary complex.

* In June 2011 an international team of scientists led by Professor So Iwata published an article in the Journal Nature detailing how using Diamond they had successfully solved the complex 3D structure of the human Histamine H1 receptor protein.

So, the arrangement of notes on the Hugh Tracey kalimba ( and on virtually any kalimba you find, as this note layout scheme has been adopted by virtually everyone who copies the Hugh Tracey kalimba ) makes some complex musical operations very simple and easy.

So only the dominant party of an immune complex is detectable in blood sera.

So these variable became what we know as Ashtekar variables, a particular flavor of Einstein-Cartan theory with a complex connection. General relativity theory expressed in this way, made possible to pursue quantization of it using well-known techniques from quantum gauge field theory.

So knowing the irreducible representations on complex spaces and their Schur indicators allows one to read off the irreducible representations on real spaces.

So many were the twists and turns in the leadership in France, and so complex were events in Saint-Domingue, that various classes and parties changed their alignments many times.

So a single complex dimension is really two dimensions, but of different kinds.

So if V < sub > J </ sub > is has complex dimension n ( real dimension 2n ) then

Guitarist Richards occasionally took a turn on bass guitar or keyboard and Klahr also incorporated keyboards, as well as a baritone guitar, though they left bass instruments out of some songs altogether, and the album is instead driven by ever-more complex and propulsive dance beats, razor sharp guitar licks (" So Many Animal Calls ", " When the Lines Go Down "), and pronounced use of multiple overlapping vocals from each member of the band (" Snow Patterns ", " No Damage Nocturne ").

So, in the 1960s a complex research of mountain Siberian pine forests of Siberia was finished by development of “ Instructions ” on forest management in them.

So elements of the Borre style are seen, for example in the " ring-chain " interlace on the Gosforth Cross, and then the complex animals of the Jelling style are mostly rather incompetently depicted in England, but traces of the next Mammen style are hard to detect ; they are much clearer on the Isle of Man.

So far only a few social species have been described, but many scientists think it is likely that there are more species in the genus with complex social behavior that have not yet been found.

So the history of the complex grains in the early solar system is complicated and only partially understood.

So large numbers of Dutch traders / pirates joined their English and French brethren on the Spanish main.

So large numbers of Dutch traders / pirates joined their English and French brethren trading on the remote coasts of Hispaniola.

So a list holding the numbers 1, 2, and 3, usually written as "( 1 2 3 )", could be represented as follows:

Long drawn-out jams in numbers like " Spoonful ", " N. S. U. ", " I'm So Glad ", and " Sweet Wine " became live favourites, while songs like " Sunshine of Your Love ", " Crossroads ", and " Tales of Brave Ulysses " remained reasonably short.

So these square roots are of positive numbers.

So, for instance, there are nonstandard models of Peano arithmetic with uncountably many ' natural numbers '.

Although Lester and Fuqua are credited as forming a spinoff group called the Moonlighters, recording in 1955 for the Chess subsidiary label Checker, they paired on only two numbers released as by the Moonlighters, " So All Alone " and " New Gal.

So far as I am aware, however, no attempt has been made to show that twins are sufficiently alike to be regarded as identical really exist in sufficient numbers to explain the proportion of twins of like sex.

Following the success of the album, the band re-released singles " Hate to Say I Told You So " and " Main Offender " which reach numbers No. 23 and No. 24 respectively in the UK Singles chart.

So far, the project has found prime numbers in eleven of the sequences, and is continuing to search in the remaining six.

So in the sense we explained above, these pairs constitute an algebra something like the real numbers.

So a more efficient method is to test if n is divisible by 2 or 3, then to check through all the numbers of form 6k ± 1.

So all prime numbers are of the form 30k + i for i = 1, 7, 11, 13, 17, 19, 23, 29 ( i. e. for i < 30 such that gcd ( i, 30 ) = 1 ).

So it was like painting by numbers ... used to love me and Percy ; we'd go in and run through our dictionary licks and he'd record them and make a loop of them.

So great was the shame, and the ill luck thought to adhere to the numbers of the Legions, that XVII, XVIII and XIX never again appear in the Roman Army's order of battle.

So we only need to know how to assign " numbers " to the elements of m < sub > x </ sub >/ m < sub > x </ sub >< sup > 2 </ sup >, and this is what the dual space does.

So the correct fourth are i and − i, which are the imaginary numbers defined to be.

So cannot be written as the sum of a countable set of countable cardinal numbers, and is regular.

So binary numbers are " base 2 "; octal numbers are " base 8 "; decimal numbers are " base 10 "; and so on.

So, for example, numbers 16 – 19 are courantes in C major, numbers 20 – 25 are sarabandes in C major, etc.