The convergence of the Rayleigh-Ritz Method in quantum chemistry

Two sufficient criteria for the convergence of the Rayleigh-Ritz Method (RRM) with respect to the eigenvalues (E-convergence) of non-relativistic electronic Hamiltonians of molecules are discussed and compared. Moreover, a necessary and sufficient criterion is given. By example (Sect. 9) it is shown that the L ²-completeness of the basis is not sufficient to guarantee E-convergence. The convergence of the wave functions in different norms (-convergence) is also investigated. In particular, sufficient conditions for the one-particle basis functions (orbitals) are given, such that a CI calculation in this basis is both E- and -convergent.     

Nowhere monotone functions and functions of nonmonotonic type

We investigate the relationships between the notions of a continuous function being monotone on no interval, monotone at no point, of monotonic type on no interval, and of monotonic type at no point. In particular, we characterize the set of all points at which a function that has one of the weaker properties fails to have one of the stronger properties. A theorem of Garg about level sets of continuous, nowhere monotone functions is strengthened by placing control on the location in the domain where the level sets are large. It is shown that every continuous function that is of monotonic type on no interval has large intersection with every function in some second category set in each of the spaces , and .

     

O-2-transitive automorphism groups with abelian stabilizer of a point

It is proved that an o-2-transitive group of order automorphisms of a totally ordered set with Abelian stabilizer of a point is the permutation groupF={φ(a, b)‖a, bεP, a>0, (x)φ(a, b)=xa+b forxεP} of a totally ordered fieldP. Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 289–293, February, 1999.     

The difference between common knowledge of formulas and sets

Common knowledge can be defined in at least two ways: syntactically as the common knowledge of a set of formulas or semantically, as the meet of the knowledge partitions of the agents. In the multi-agent S5 logic with either finitely or countably many agents and primitive propositions, the semantic definition is the finer one. For every subset of formulas that can be held in common knowledge, there is either only one member or uncountably many members of the meet partition with this subset of formulas held in common knowledge. If there are at least two agents, there are uncountably many members of the meet partition where only the tautologies of the multi-agent S5 logic are held in common knowledge. Whether or not a member of the meet partition is the only one corresponding to a set of formulas held in common knowledge has radical implications for its topological and combinatorial structure.     

Second-Order Optimality Conditions in Multiobjective Optimization Problems

In this paper, we develop second-order necessary and sufficient optimality conditions for multiobjective optimization problems with both equality and inequality constraints. First, we generalize the Lin fundamental theorem (Ref. 1) to second-order tangent sets; then, based on the above generalized theorem, we derive second-order necessary and sufficient conditions for efficiency.     

Collapse of the magnetic ordering and structural anomalies in the U La S system: neutron diffraction and specific heat measurements

We report specific heat and neutron diffraction measurements of seven samples in the solid solution system U_xLa_1-xS. All samples have the simple fcc NaCl crystal structure. Both specific heat and neutron diffraction confirm the suggestion from the earlier magnetic measurements that the ferromagnetism disappears abruptly at 0.57. Near there is a doubling of the electronic contribution to the specific heat, as compared to the value of 23 mJ mol^-1K^-2 in pure US. Around the widths of the nuclear Bragg peaks show a considerable broadening, as well as anomalies in the mean lattice parameter, as compared to those expected from Vegard's law. A preliminary analysis suggests this broadening may be due to a loss of long range lattice order near . However, these changes are independent of temperature, so that further experiments are necessary before they can be associated with the changes in magnetic behavior at . Received 18 September 1998     

Ga-NMR study of the low-energy excitations in

We report the results of ⁶⁹Ga- and ⁷¹Ga-NMR measurements on NdGa₂ at temperatures between 0.1 and and in applied magnetic fields between zero and 74 kOe. NdGa₂ orders antiferromagnetically below and undergoes several metamagnetic transitions in external magnetic fields. In zero applied magnetic field and below the temperature dependence of the spin-lattice relaxation rate T1 ^-1 ( T ) shows a large linear-in-T term, about two orders of magnitude higher than for the reference compound LaGa₂. This strong enhancement confirms the presence of low-energy excitations in the antiferromagnetic phase of NdGa₂ as was previously indicated by specific heat data. Above , T1 ^-1 ( T ) is dominated by an exponential term, which we associate with excitations between the lowest energy levels of the f-electron system. The separation of these energy levels is determined by exchange, crystal-field and Zeeman interactions. Received 3 September 1998 and Received in final form 3 November 1998     

Extension of Simons' inequality

We prove the following extended version of Simons' inequality and present its applications. Let be a set and be a subset of . Let be a subset of a Hausdorff topological vector space which is invariant under infinite convex combinations. Let be a bounded function such that the functions are convex for all and whenever 0$">, and Let be a sequence in . Assume that, for every , there exists satisfying . Then

If , then the set in the above inequality can be replaced by .

     

Even-tempered Slater-type orbitals revisited: from hydrogen to krypton

Even-tempered Slater-type orbital basis sets were developed in 1973, based on total atomic energy optimization. Here, we revisit ET STOs and propose new sets based on past experience and recent computational studies. From preliminary atomic and molecular tests, these sets are shown to be very well balanced and to perform, at lower cost, almost as well as a very large (close to complete) basis set.     

Uniform non-amenability

For any finitely generated group G an invariant ?0 is introduced which measures the “amount of non-amenability” of G. If G is amenable, then . If , we call G uniformly non-amenable. We study the basic properties of this invariant; for example, its behaviour when passing to subgroups and quotients of G. We prove that the following classes of groups are uniformly non-amenable: non-abelian free groups, non-elementary word-hyperbolic groups, large groups, free Burnside groups of large enough odd exponent, and groups acting acylindrically on a tree. Uniform non-amenability implies uniform exponential growth. We also exhibit a family of non-amenable groups (in particular including all non-solvable Baumslag-Solitar groups) which are not uniformly non-amenable, that is, they satisfy . Finally, we derive a relation between our uniform Følner constant and the uniform Kazhdan constant with respect to the left regular representation of G.