Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.

Indeed, MSK is a particular case of the sub-family of CPM known as continuous-phase frequency-shift keying ( CPFSK ) which is defined by a rectangular frequency pulse ( i. e. a linearly increasing phase pulse ) of one symbol-time duration ( total response signaling ).

Indeed, the transformation of Newfoundland English offers a case study of the politics of language.

Indeed, in the latter case, it is only defined for non-negative integer arguments.

For the univariate case, Indeed, for random variables X and Y, the expectation of their product is an inner product.

Indeed, as Kotowicz notes, his attention was drawn more to the case presented by Richard Brickner, at the same conference, of a patient who had had his frontal lobes ablated and, while experiencing a flattening of affect, had suffered no apparent decrease in intellect.

Indeed, it is false, because it is not the case that there exists a unique individual who is currently the king of France and is bald — since there is no present king of France ( Bertolet 1999 ).

Indeed, Chief Justice William Rehnquist repeatedly criticized the Court's invention of corporate constitutional " rights ," most famously in his dissenting opinion in the 1978 case First National Bank of Boston v. Bellotti.

Indeed, there are non-Bayesian updating rules that also avoid Dutch books ( as discussed in the literature on " probability kinematics " following the publication of Richard C. Jeffrey's rule, which applies Bayes ' rule to the case where the evidence itself is assigned a probability ).

Indeed, he worked alone ( which is rather rare for a robber ) and carried his machine gun in a violin case.

Indeed several agencies and experts and journals have published concerns about viewing the developments as a case of genocide: Roméo Dallaire, Genocide Watch, Sentinel Project for Genocide Prevention, and the journals War Crimes, Genocide, & Crimes against Humanity and Journal of Genocide Research.

Indeed, one of the common markers of a case of SHC is that the body – or part of it – suffered an extraordinarily large degree of burning while the surroundings or the lower limbs remained comparatively undamaged.

Indeed the approval has little value in any case: it should be noted the subject of the article is not the classic declaration of war, as such a declaration according to doctrine might constitute a war crime by implying a war of aggression forbidden by international law.

Indeed, Fox's generous nature ( helped by his earlier over-indulgence by his father ) made him a rather hopeless political party-leader, often seeming quite unwilling to cajole or discipline his unruly supporters, especially in the case of the young, who were disproportionately represented among the Foxites.

" Indeed defenders of EMH maintain that Behavioral Finance strengthens the case for EMH in that BF highlights biases in individuals and committees and not competitive markets.

Indeed, if that is not the case to start with, a different set of intervals can be picked for which these assumptions hold.

Indeed, it is very frequently the case that one can even predict and explain the kind of mental or psychological deterioration or change that human beings will undergo when specific parts of their brains are damaged.

Indeed, the average intercarrier compensation rate ranges from 0. 1 cents per minute for traffic bound to an information service provider (“ ISP ”) to 5. 1 cents per minute for intrastate traffic bound to a subscriber of a small ( rural ) incumbent local exchange carrier ; individual rates can be as low as zero and as high as 35. 9 cents per minute — even though in each case basically the same transport and switching functions are provided.

An important consequence of the domain invariance theorem is that R < sup > n </ sup > cannot be homeomorphic to R < sup > m </ sup > if m ≠ n. Indeed, no non-empty open subset of R < sup > n </ sup > can be homeomorphic to any open subset of R < sup > m </ sup > in this case.

Indeed, at least one case has been documented of the pro-Nazi and white supremacist Liberty Lobby distributing a pro-Wallace pamphlet entitled " Stand up for America " despite the campaign's denial of such a connection.

Indeed, in this case we introduce the unitary transformation

Indeed, in the case of Kirmani v Captain Cook Cruises Pty Ltd ( No 2 ) ( 1985 ), the court said that it would never again grant a certificate of appeal.

Indeed, the common metaphor for the strongest possible evidence in any case — the " smoking gun "— is an example of proof based on circumstantial evidence.

Indeed, it may be the case that Roméo et Juliettes genesis is intertwined with other works composed before the composer left for his Prix de Rome sojourn of 1830-32.

Indeed, the area of a disk of radius R is πR < sup > 2 </ sup > and the circumference of the circle is 2πR, so both sides of the inequality are equal to 4π < sup > 2 </ sup > R < sup > 2 </ sup > in this case.

Indeed, the effect was quite the contrary, psychology began to thrive: " perhaps, in the early days of scientific progress, a subject often grows all the more surely if its workers have to meet difficulties, improvise their apparatus, and rub very close shoulders one with another.

Former Senate Prayer Group member and current Kansas Governor Sam Brownback has described group members ' method of operation: “ Typically, one person grows desirous of pursuing an action ”—- a piece of legislation, a diplomatic strategy —-“ and the others pull in behind .” Indeed, Brownback has often joined with fellow Family members in pursuing legislation.

Indeed, the probability that the least mutated genomes in an asexual population end up carrying at least one ( additional ) mutation depends heavily on the genomic mutation rate and this increases more or less linearly with the size of the genome ( more accurately, with the number of base pairs present in active genes ).

Indeed, if pressed, we would say what the late Robert Henri, American painter, said to a pupil, `` Anything will do for a subject: it's what you do with it that counts ''.

" Indeed, the song " Do You Believe In Magic ," a Top Ten hit, mentioned the genre in its lyrics: " If you believe in magic, don't bother to choose / If it's jug band music or rhythm and blues / Just go and listen, it'll start with a smile / That won't wipe off your face no matter how hard you try.

: Cassilda: Indeed it's time.

" Indeed, on the importance of astronomical knowledge, Faber states that " it's astronomy that puts us in perspective ; it tells us where we come from, and cosmically, where we're going.

Indeed, it would be wrong to say that STX is a general purpose transformation language ; however, if your transformation needs can be met by STX then it's an efficient and smart choice.

Indeed, one has to pay careful attention to figure out just what it is that who is doing to whom in Three Days of the Condor and, if I understood it correctly, it's never as horrifying as the real thing ... The suspense of the film depends less on this kind of plausibility than on Mr. Redford's reputation ( in a movie we accept the fact that he can do anything ) and on the verve with which Mr. Pollack, the director, sets everything up.

" Indeed, George Sand noted, after Deburau's death, that the " titis ", the street boys, of the Funambules seemed to regard his Pierrot as their " model "; but, earlier, when she had asked Deburau himself what he thought of Janin's conclusions, he had had this to say: " the effect is of service to my reputation, but all that is not the art, it's not the idea I have of it.

Indeed, there were Siamese performers in the royal court of Cambodia during the 19th century according to most renown sources on the royal ballet, Groslier included ; this suggests a strong connection to the court dances of Siam and it's influences.

Indeed, it coincides with the topology of pointwise convergence of linear functionals.

Indeed, this follows since just as the image of a vector subspace under a linear transformation is a subspace, so is the image of V under L. But the only subspaces ( i. e., k-subspaces ) of k are

Indeed what was done is almost precisely to explain that tensor spaces are the constructions required to reduce multilinear problems to linear problems.

Indeed, fractional coloring problems are much more amenable to a linear programming approach than traditional coloring problems.

Indeed, area coordinates will also allow us to perform a linear interpolation at all points in the triangle if the values of the function are known at the vertices.

Indeed, if f: U → X is such a function and T: X → C a bounded linear functional, one can show that

Indeed, a linear, time-invariant system ( see LTI system theory ) is said to be BIBO stable if and only if bounded inputs produce bounded outputs ; this is equivalent to requiring that the denominator of its transfer function ( which can be proven to be rational ) is stable.