The maximal monotonicity of the subdifferentials of convex functions: Simons' problem

In the paper we deal with the problem when the graph of the subdifferential operator of a convex lower semicontinuous function has a common point with the product of two convex nonempty weak and weak^* compact sets, i.e. when graph (Q × Q ^*) 0. The results obtained partially solve the problem posed by Simons as well as generalize the Rockafellar Maximal Monotonicity Theorem.     

Inclusion-exclusion: Exact and approximate

It is often required to find the probability of the union of givenn eventsA ₁ ,...,A _n . The answer is provided, of course, by the inclusion-exclusion formula: Pr(A _i )= _i – _i<j Pr(A _i A _j )±.... Unfortunately, this formula has exponentially many terms, and only rarely does one manage to carry out the exact calculation. From a computational point of view, finding the probability of the union is an intractable, #P-hard problem, even in very restricted cases. This state of affairs makes it reasonable to seek approximate solutions that are computationally feasible. Attempts to find such approximate solutions have a long history starting already with Boole [1]. A recent step in this direction was taken by Linial and Nisan [4] who sought approximations to the probability of the union, given the probabilities of allj-wise intersections of the events forj=1,...k. The developed a method to approximate Pr(A _i ), from the above data with an additive error of exp . In the present article we develop an expression that can be computed in polynomial time, that, given the sums _|S|=j Pr( _iS A _i ) forj=1,...k, approximates Pr(A _i ) with an additive error of exp . This error is optimal, up to the logarithmic factor implicit in the notation.The problem of enumerating satisfying assignments of a boolean formula in DNF formF=v _l ^m C _i is an instance of the general problem that had been extensively studied [7]. HereA _i is the set of assignments that satisfyC _i , and Pr( _iS A _i )=a _S /2ⁿ where _iS C _i is satisfied bya _S assignments. Judging from the general results, it is hard to expect a decent approximation ofF's number of satisfying assignments, without knowledge of the numbersa _S for, say, all cardinalities . Quite surprisingly, already the numbersa _S over |S|log(n+1)uniquely determine the number of satisfying assignments for F.We point out a connection between our work and the edge-reconstruction conjecture. Finally we discuss other special instances of the problem, e.g., computing permanents of 0,1 matrices, evaluating chromatic polynomials of graphs and for families of events whose VC dimension is bounded.Work supported in part by a grant of the Binational Israel-US Science Foundation.Work supported in part by a grant of the Binational Israel-US Science Foundation and by the Israel Science Foundation.     

Crystal and molecular structure of 6-0-acetylgrayanotoxin III

The crystal and molecular structure of a grayanotoxin III derivative, 6-0-acetylgrayanotoxin III is presented. The crystal is orthorhombic, space group P2₁2₁2₁, witha=15.582(3),b=21.304(3),c=6.339(3)Å,V=2104(1) ų Z=4. The structure was solved by direct methods and refined by full matrix least-squares methods to a finalR=0.047 for 1763 independent reflections withF ₀>3 (F ₀) The molecule is based on a tetracyclic structure consisting of two five-membered, one six-membered, and one seven-membered ring with various conformations.     

An inequality relating algebraic and trigonometric derivatives

Using the correspondence x↔ cos θ, where −1≤x ≤ 1 and 0 ≤ θ ≤ π, a function f(x) defined on [−1, 1] can be represented as a 2π-periodic function F(θ), and then the derivative f′(x) corresponds to . From these observations, weighted-norm estimates for first and higher derivatives by x will be obtained, using a generalized Hardy inequality. The results in turn imply the generalized Hardy inequality upon which they depend and will hold true in any weighted norm for which the generalized Hardy is true.     

Behavior of alternation points in best rational approximation

The behavior of the equioscillation points (alternants) for the error in best uniform approximation on [–1, 1] by rational functions of degreen is investigated. In general, the points of the alternants need not be dense in [–1, 1], even when approximation by rational functions of degree (m, n) is considered and asymptoticallym/n 1. We show, however, that if more thanO(logn) poles of the approximants stay at a positive distance from [–1, 1], then asymptotic denseness holds, at least for a subsequence. Furthermore, we obtain stronger distribution results when n (0 < 1) poles stay away from [–1, 1]. In the special case when a Markoff function is approximated, the distribution of the equioscillation points is related to the asymptotics for the degree of approximation.The research of this author was supported, in part, by NSF grant DMS 920-3659.     

On the Landau-Teller approximation for energy exchanges with fast degrees of freedom

We revisit the Landau-Teller heuristic approach to adiabatic invariants and, following Rapp, use it to investigate the energy exchanges between the different degrees of freedom, in simple Hamiltonian systems describing the collision of fast rotating or vibrating molecules with a fixed wall. We critically compare the theoretical results with particularly accurate numerical computations (quite small energy exchanges, namely of one part over 10³⁰, are measured).     

Asymptotics for the remainder of a class of positive quadratures for integrands with an interior singularity

Summary For functions with an interior singularity, the errors of a class of positive quadrature formulae with high algebraic degree are reduced to those of the much simpler Euler-Maclaurin type formulae. Applying this method to certain classes of functions, such as, for example,f(x)=h(x)|x-u| , where >–1, with a sufficiently smooth functionh, we obtain the main term of the error expansion for quadrature rules of ultraspherical type.     

Effect of Hydration–Dehydration Processes on the Structure and Porosity of Detemplated Silica Fibers

A study was carried out on the conditions of template removal (calcination or extraction by solvents) and hydration–dehydration processes on the hexagonal structure and porosity of detemplated fibers. The hydration–dehydration cycle leads to a decrease in fiber porosity, especially in fiber detemplation by extraction with surfactant solvents such as ethanol and acetic acid.     

An analysis of the complex molecular orbitals method

Numerical results for the ground state of the HN ₂ ⁺ and HCO⁺ molecular ions at their near equilibrium geometry, obtained by the complex molecular orbitals (CMO) method in the extended basis set, are reported. The CMO wavefunction of the HN ₂ ⁺ ion is compared with the CI wavefunction obtained in the same basis set. This reveals the nature of approximations inherent in the CMO method. A peculiar feature of the occupation numbers of the CMO natural orbitals is also explained.Alexander von Humboldt Fellow. On leave from the Institute Rudjer Bokovi, Zagreb, Croatia, Yugoslavia.     

Surface tension measurements in the study of molecular complexes

Surface tension measurements can be used to investigate molecular complex formation in liquid solutions for strong and weak complexes. The association constant and epthalpy for triethylamine-iodine, hexamethylbenzene-tetracyanoethylene and ethanol-iodine in cyclohexane are 4·55×10³, 218 and 0.93M⁻¹ at 25° C and 12·5, 7·7 and 5·1 kcal/mol respectively. These values compare well with those reported in the literature by other methods.