Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

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Methods Of Modern Mathematical Physics

Convex bodies: the brunn–minkowski theory

5. M. Green, J. Schwarz, and E. Witten, Superstring theory, Cambridge Univ. Press, 1987. 6. J. Isenberg, P. Yasskin, and P. Green, Non-self-dual gauge fields, Phys. Lett. 78B (1978), 462-464. 7. B. Kostant, Graded manifolds, graded Lie theory, and prequantization, Differential Geometric Methods in Mathematicas Physics, Lecture Notes in Math., vol. 570, SpringerVerlag, Berlin and New York, 1977. 8. C. LeBrun, Thickenings and gauge fields, Class. Quantum Grav. 3 (1986), 1039-1059. 9. , Thickenings and conformai gravity, preprint, 1989. 10. C. LeBrun and M. Rothstein, Moduli of super Riemann surfaces, Commun. Math. Phys. 117(1988), 159-176. 11. Y. Manin, Critical dimensions of string theories and the dualizing sheaf on the moduli space of (super) curves, Funct. Anal. Appl. 20 (1987), 244-245. 12. R. Penrose and W. Rindler, Spinors and space-time, V.2, spinor and twistor methods in space-time geometry, Cambridge Univ. Press, 1986. 13. R. Ward, On self-dual gauge fields, Phys. Lett. 61A (1977), 81-82. 14. E. Witten, An interpretation of classical Yang-Mills theory, Phys. Lett. 77NB (1978), 394-398. 15. , Twistor-like transform in ten dimensions, Nucl. Phys. B266 (1986), 245-264. 16. , Physics and geometry, Proc. Internat. Congr. Math., Berkeley, 1986, pp. 267302, Amer. Math. Soc, Providence, R.I., 1987.

Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Roger Temam)

Random matrices now play a role in many areas of theoretical, applied,and computational mathematics. Therefore, it is desirable to have toolsfor studying random matrices that are flexible, easy to use, and powerful.Over the last fifteen years, researchers have developed a remarkablefamily of results, called matrix concentration inequalities, that achieveall of these goals.This monograph offers an invitation to the field of matrix concentrationinequalities. It begins with some history of random matrix theory;it describes a flexible model for random matrices that is suitablefor many problems; and it discusses the most important matrix concentrationresults. To demonstrate the value of these techniques, thepresentation includes examples drawn from statistics, machine learning,optimization, combinatorics, algorithms, scientific computing, andbeyond.

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An Introduction to Matrix Concentration Inequalities

A compact, physical, short-channel threshold voltage model for undoped symmetric double-gate MOSFETs has been derived based on an analytical solution of the two-dimensional (2-D) Poisson equation with the mobile charge term included. The new model is verified by published numerical simulations with close agreement. Applying the newly developed model, threshold voltage sensitivities to channel length, channel thickness, and gate oxide thickness have been comprehensively investigated. For practical device designs the channel length causes 30-50% more threshold voltage variation than does the channel thickness for the same process tolerance, while the gate oxide thickness causes the least, relatively insignificant threshold voltage variation. Model predictions indicate that individual DG MOSFETs with good turn-off behavior are feasible at 10 nm scale; however, practical exploitation of these devices toward gigascale integrated systems requires development of novel technologies for significant improvement in process control.

A physical short-channel threshold voltage model for undoped symmetric double-gate MOSFETs

The third volume of this textbook on mathematical physics is devoted to quantum mechanics and especially to its applications to atomic systems. The author builds the theory on an axiomatic basis and uses the functional analysis of unbounded operators to bridge the gap to physics and its experiments. Mathematics is shown to explain the axioms in depth and to provide the right tool for testing numerical data in experiments. The topics covered are Hilbert spaces, the theory of unbounded operators, scattering theory, atoms and molecules.

Quantum mechanics of atoms and molecules

In this article, we prove and exploit a trace identity for the spectra of Schrodinger operators and similar operators. This identity leads to universal bounds on the spectra, which apply to low-lying eigenvalues, eigenvalue asymptotics, and to partition functions (traces of heat operators). In many cases they are sharp in the sense that there are specific examples for which the inequalities are saturated. Special cases corresponding to known inequalities include those of Hile and Protter and of Yang. Introduction In this article, we prove and exploit an identity for the spectra of self-adjoint operators H modeled on the Dirichlet Laplacian or, more generally, on Schrodinger operators of the form (p−A(x))2 + V (x), (1) where p = 1i ∇ is the usual momentum operator in convenient units and A(x) is the magnetic vector potential. We recover and extend several known inequalities involving sums, differences, and ratios of eigenvalues. Let λj , j = 1, . . . , denote the ordered eigenvalues of the Dirichlet Laplacian on a bounded d-dimensional domain with zero Dirichlet boundary conditions, and recall that Hile and Protter [HiPr80] proved that: d 4 ≤ 1 n n ∑ j=1 λj λn+1 − λj , (2) thereby extending an earlier inequality of Payne, Polya, and Weinberger [PaPoWe56]. In the last few years it has become clear that these and many similar relationships can be realized as special cases of abstract variational bounds involving the interplay among commutators of −∇2, a Cartesian coordinate xj , and the corresponding derivative ∂/∂xj. For this analysis and various extensions of (2) see [Ha88], [Ho90], [Ha93], [HaMi95]. While inequalities of this type have been fairly sharp for low-lying eigenvalues, they have been mostly disappointing for higher eigenvalues. Yang [Ya91], however, Received by the editors September 28, 1995. 1991 Mathematics Subject Classification. Primary 35J10, 35J25, 58G25.

https://www.ams.org/tran/1997-349-05/S0002-9947-97-01846-1/S0002-9947-97-01846-1.pdf

On trace identities and universal eigenvalue estimates for some partial differential operators

Quantum Mechanics of Large Systems

It is shown how the rigorous justification of resonance widths in Paper I [5] can be simplified by exploiting Langer's trick of expanding the independent variable rather than the dependent variable [9].

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Evans M. Harrell

Papers

A physical short-channel threshold voltage model for undoped symmetric double-gate MOSFETs

Quantum mechanics of atoms and molecules

On trace identities and universal eigenvalue estimates for some partial differential operators

Quantum Mechanics of Large Systems

The mathematical theory of resonances whose widths are exponentially small