Birkhoff

It has become fashionable to talk about density functional theory, so we had better come to grips with the term linear functional. It is a highly specialized term in modern algebra with the following meaning (for the mathematicians amongst you, and paraphrased from Birkhoff and MacLane (1965 182)) ... 

Original reference G.D.Birkhoff, Nouvellcs Recherches Snr les Systems Dynamiques , Mem. Pont, Acad. Set. Novi Lyncaei 1 (1935) 85. 

ThenWs(gfe( j) and Wu 7e(t)) intersect transversely at(xh(rio)+0(e),9 j and consequently (from, theS male-Birkhoff homoclinic theorem) for the map P there exists an integer n > 1 that P has an invariant Cantor set on which it is topologically conjugate to a full shift of N symbols. 

S. Mac Lane, G. Birkhoff, Algebra, Macmillan Company, New York, 1967, p. 3 Given sets of R and S have the intersection R H 5 with the common elements R and... 

Birkhoff, R. and Nurminen, M. (2004) NExOCTANE Process for isooctane production, in Handbook of Petroleum Refining Processes, 3nd edn (ed. R.A. Myers), McGraw-Hill, pp. 1.3-1.9. 

Finite-additive invariant measures on non-compact groups were studied by Birkhoff (1936) (see also the book of Hewitt and Ross, 1963, Chapter 4). The frequency-based Mises approach to probability theory foundations (von Mises, 1964), as well as logical foundations of probability by Carnap (1950) do not need cr-additivity. Non-Kolmogorov probability theories are discussed now in the context of quantum physics (Khrennikov, 2002), nonstandard analysis (Loeb, 1975) and many other problems (and we do not pretend provide here is a full review of related works). 

Theory of Jet Formation in Lined Shaped Charges. See Detonation BMPT (Birkhoff-MacDougall-Pugh-Taylor) Theory of Jet Formation in Shaped Charges... 

Munroe-Neumann effect is the property of explosive charges detonated in contact with objects of steel, concrete etc, to produce much deeper holes in the objects when the charge contains a cavity than when the explosive is solid. According to Birkhoff et al, (Ref 28) this property of explosives has been known for at least 150 years, the earliest reference to it being in 1792. 

The basic principle of the lined cavity phenomenon is as follows, as revealed by high-speed radiographs and described by Birkhoff et al (Ref 28) ... 

Birkhoff et al (Ref 28) gave a fairly complete mathematical theory of cavity-effect phenomenon together with experimental data that aided the formulation and testing of the theory. They based the theory upon the classical thermodynamics of perfect fluids. It is applicable because the strength of metals used for linings can be neglected at the high pressures encountered... 

The hydrodynamic theory of the penetration of targets by lined cavity jets was developed, according to Cook (Ref 7, p 252), independently by Pugh (Birkhoff s et al Ref 2) and by Hill et al (Ref 1). Pack Evans (Refs 3 4) discussed the steady-state theory of penetration in which the jet-velocity. distribution was ignored and the penetration velocity was assumed constant. Pugh, 8t Eichelberger (Refs 5 6) discussed the nonsteady-state of jet penetration in which, the actual velocity distribution in the jet was taken into account as well as the variation of the velocity of penetration with.depth These theories are discussed by Cook (Ref 7) Refs 1) R. Hill, N.F. Mott D.C. Pack, Unpublished "Ministry of Supply Report, January 1944 2) G. Birkhoff, D.P. Mac-... 

Birkhoff-MacDougall-Pugh-Taylor Theory of Jet Formation. See next item... 

Jet Formation in Shaped Charges. See Detonation BMT (Birkhoff-MacDougall-Pugh-Taylor) Theory of Jet Formation in Shaped Charges in Vol 4 of Encycl, p D226-R and Detonation, Munroe-Neumann Effect (or Shaped. Charge Effect) and Lined-Cavity Effect in on pp D442-Rff... 

G. Birkhoff et al, JAppliedPhys 19, 563-82(1948) CA 42, 8475(1948) (Explosives with lined cavities — mathematical theory of lining with metal the "Munroe-effect cavity to increase armor pierving capability)... 

Vol. 6. Numerical Solution of Elliptic Problems Garret Birkhoff and Robert E. Lynch... 

Detonation BMPT (Birkhoff-Macdougall-Pough-Taylor) theory of jet formation in shaped charges 4 D226... 

This is a simple analogy of Birkhoff s ergodic theorem for dynamical systems, see A.I. Khinchin, Mathematical Foundation of Statistical Mechanics (Dover, New York 1949) L.E. Reichl, A Modern Course in Statistical Physics (University of Texas Press, Austin, TX 1980) ch. 8. 

The anharmonicities of the potential contribute by the terms involving the constants x, g, y,. .. as well as the energy shifts AEx = 0(h2),. .. and the frequency shifts Aw, = 0(h2),. These anharmonic constants can be calculated by the Van Vleck contact transformations [20] as well as by a semi-classical method based on an h expansion around the equilibrium point [14], which confirms that the Dunham expansion (2.8) is a series in powers of h. Systematic methods have been developed to carry out the Van Vleck contact transformations, as in the algebraic quantization technique by Ezra and Fried [21]. It should be noted that the constants x and g can also be obtained from the classical-mechanical Birkhoff normal forms [22], The energy shifts AEx,... 

This result was rederived shortly afterwards by Birkhoff, et al. (B7), and by Scriven (S4), in considering the growth of gas or vapor bubbles. Finally, from Eqs. (24), (31), and (32), it follows that... 

The word list has been adopted, following Birkhoff Maclane (Algebra. New York. Macmillan 1967, p. viii), in place of the ugly word n-tuple. It will usually be quite clear how many elements are involved in case of doubt we shall refer to a list of n elements. 

G. D. Birkhoff tried to give a mathematical theory of aesthetics by recognizing the complexity, C, and the order, O (i.e., its harmony and symmetry) that an object could be said to have. He then restated Hemsterhuis definition of the beautiful as that which gives us the greatest number of ideas in the shortest time as the relation aesthetic measure... 

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