In general the position as regards all such new calculi is this - That one cannot accomplish by them anything that could not be accomplished without them. However, the advantage is, that, provided such a calculus corresponds to the inmost nature of frequent needs, anyone who masters it thoroughly is able - without the unconscious inspiration of genius which no one can command - to solve the respective problems, yea to solve them mechanically in complicated cases in which, without such aid, even genius becomes powerless. Such is the case with the invention of general algebra, with the differential calculus, and in a more limited region with Lagrange's calculus of variations, with my calculus of congruences, and with Mobius's calculus. Such conceptions unite, as it were, into an organic whole countless problems which otherwise would remain isolated and require for their separate solution more or less application of inventive genius.
Carl Friedrich GaussRelated topics
able advantage algebra anyone anything calculus case command differential general genius inmost less limited nature needs organic position region remain unconscious yea mastersRelated quotes
It has been said, that the people had already surrendered all their powers to the State sovereignties, and had nothing more to give. But, surely, the question whether they may resume and modify the powers granted to government does not remain to be settled in this country. Much more might the legitimacy of the general government be doubted, had it been created by the States. The powers delegated to the State sovereignties were to be exercised by themselves, not by a distinct and independent sovereignty, created by themselves. To the formation of a league, such as was the confederation, the State sovereignties were certainly competent. But when, "in order to form a more perfect union," it was deemed necessary to change this alliance into an effective government, possessing great and sovereign powers, and acting directly on the people, the necessity of 'referring it to the people, and of deriving its powers directly from them, was felt and acknowledged by all.
John Marshall
The American people are right in demanding that New Nationalism, without which we cannot hope to deal with new problems. The New Nationalism puts the national need before sectional or personal advantage. It is impatient of the utter confusion that results from local legislatures attempting to treat national issues as local issues. It is still more impatient of the impotence which springs from over division of governmental powers, the impotence which makes it possible for local selfishness or for legal cunning, hired by wealthy special interests, to bring national activities to a deadlock. This New Nationalism regards the executive power as the steward of the public welfare. It demands of the judiciary that it shall be interested primarily in human welfare rather than in property, just as it demands that the representative body shall represent all the people rather than any one class or section of the people.
Theodore Roosevelt
Men have looked upon the desert as barren land, the free holding of whoever chose; but in fact each hill and valley in it had a man who was its acknowledged owner and would quickly assert the right of his family or clan to it, against aggression. Even the wells and trees had their masters, who allowed men to make firewood of the one and drink of the other freely, as much as was required for their need, but who would instantly check anyone trying to turn the property to account and to exploit it or its products among others for private benefit. The desert was held in a crazed communism by which Nature and the elements were for the free use of every known friendly person for his own purposes and no more. Logical outcomes were the reduction of this licence to privilege by the men of the desert, and their hardness to strangers unprovided with introduction or guarantee, since the common security lay in the common responsibility of kinsmen.
T. E. Lawrence
A nonlinear differential equation of the Riccati type is derived for the covariance matrix of the optimal filtering error. The solution of this 'variance equation' completely specifies the optimal filter for either finite or infinite smoothing intervals and stationary or non-stationary statistics.
The variance equation is closely related to the Hamiltonian (canonical) differential equations of the calculus of variations. Analytic solutions are available in some cases. The significance of the variance equation is illustrated by examples which duplicate, simplify, or extend earlier results in this field.
The duality principle relating stochastic estimation and deterministic control problems plays an important role in the proof of theoretical results. In several examples, the estimation problem and its dual are discussed side-by-side.
Properties of the variance equation are of great interest in the theory of adaptive systems. Some aspects of this are considered briefly.
Rudolf E. Kálmán
Dear Mr. Rosenberg - I was very surprised this morning that you refused to give me 2,000 fr. especially given that I had told you that my wife was gravely ill in Berlin and that I needed to leave immediately. You probably thought it was a tale I was telling you in order to shave some money off the sum that you owe me; to prove to you that this is not the case, I enclose with this letter a telegram that I received this morning. Strong though the crisis may be and as strongly as you may have applied the brakes, it is not possible that you are not able to find 1,000 fr. for a painter with whom you have done business for six years now and who (I dare hope) enjoys all your esteem; especially given that the said painter tells you that his wife is seriously ill and that he needs to leave.
I should tell you that in general the way you act in these times of crises is not particularly heroic.
Giorgio de Chirico