Let A ⊂ R be a set-germ at 0 ∈ R such that 0 ∈ A. We say that r ∈ S is a direction of A at 0 ∈ R if there is a sequence of points {xi} ⊂ A\{0} tending to 0 ∈ R such that xi ‖xi‖ → r as i → ∞. Let D(A) denote the set of all directions of A at 0 ∈ R. Let A, B ⊂ R be subanalytic set-germs at 0 ∈ R such that 0 ∈ A ∩ B. We study the problem of whether the dimension of the common direction set, dim(D...

The Central Sets Theorem is a powerful theorem, one of whose consequences is that any central set in N contains solutions to any partition regular system of homogeneous linear equations. Since at least one set in any finite partition of N must be central, any of the consequences of the Central Sets Theorem must be valid for any partition of N. It is a result of Beiglböck, Bergelson, Downarowicz...

In this paper we introduce a new class of sets called g̃αclosed sets in topological space. We prove that this class lies between αclosed sets and gα-closed sets.We discuss some basic properties of g̃αclosed sets. Applying this we introduce a new space called T gαspace. Mathematics Subject Classification: 54C08

Density functional theory (DFT) provides a formally exact framework for performing embedded subsystem electronic structure calculations, including DFT-in-DFT and wavefunction theory-in-DFT descriptions. In the interest of efficiency, it is desirable to truncate the atomic orbital basis set in which the subsystem calculation is performed, thus avoiding high-order scaling with respect to the size...

A two-electron basis-set completeness profile is proposed by analogy with the one-electron profile introduced by D. P. Chong (Can J Chem 1995, 73, 79). It is defined as Y(α, β) = ∑m ∑n〈Gα(1)Gβ(2)|(1/r12)| m(1) n(2)〉〈 m(1) n(2)|r12|Gα(1)Gβ(2)〉 and motivated by the expression for the basis-set truncation correction that occurs in the framework of explicitly correlated methods. The two-electron ba...

The motor cortex was the one of the first cortical areas to be explored electrophysiologically, yet little agreement has emerged regarding its basic response properties. Often it is assumed that single-neuron responses reflect a preference for a particular movement or movement variable. It may be further assumed that movement is generated by (or at least accompanied by) a growing population-lev...

We have calculated position and dot intracules for a series of atomic and molecular systems, starting from an unrestricted Hartree-Fock wave function, expanded using the STO-3G, 6-31G, 6-311G, 6-311++G, 6-311++G(d,p), 6-311++G(3d,3p), and 6-311++G(3df,3pd) basis sets as well as the nonpolarized part of Dunning's cc-pV5Z basis. We find that the basis set effects on the intracules are small and t...