The regularity of the η\eta function for the Shubin calculus

We prove the regularity of the $\eta$ function for classical pseudodifferential operators with Shubin symbols. We recall the construction of complex powers and of the Wodzicki and Kontsevich-Vishik functionals for classical symbols on $\mathbb{R}^{n}$ with these symbols. We then define the $\zeta$ and $\eta$ functions associated to suitable elliptic operators. We compute the $K_{0}$ group of the algebra of zero-order operators and use this knowledge to show that the Wodzicki trace of the idempotents in the algebra vanishes. From this, it follows that the $\eta$ function is regular at 0 for any self-adjoint elliptic operator of positive order

Removing Colors 2k, 2k-1, and k

We prove that if a link admits non-trivial (2k+1)-colorings, with prime 2k+1>7, it also admits non-trivial (2k+1)-colorings not involving colors 2k, 2k-1, nor k

Permutations Which Make Transitive Groups Primitive

In this article we look into characterizing primitive groups in the following way. Given a primitive group we single out a subset of its generators such that these generators alone (the so-called primitive generators) imply the group is primitive. The remaining generators ensure transitivity or comply with specific features of the group. We show that, other than the symmetric and alternating groups, there are infinitely many primitive groups with one primitive generator each. These primitive groups are certain Mathieu groups, certain projective general and projective special linear groups, and certain subgroups of some affine special linear groups.Comment: 10 pages; version 2 accepted for publication in CEJ

Partial profiles of quasi-complete graphs

We enumerate graph homomorphisms to quasi-complete graphs, i.e., graphs obtained from complete graphs by removing one edge. The source graphs are complete graphs, quasi-complete graphs, cycles, paths, wheels and broken wheels. These enumerations give rise to sequences of integers with two indices; one of the indices is the number of vertices of the source graph, and the other index is the number of vertices of the target graph.Comment: 21 pages, 5 figure

Hyperfinite knots via the CJKLS invariant in the thermodynamic limit

We set forth a definition of hyperfinite knots. Loosely speaking, these are limits of certain sequences of knots with increasing crossing number. These limits exist in appropriate closures of quotient spaces of knots. We give examples of hyperfinite knots. These examples stem from an application of the Thermodynamic Limit to the CJKLS invariant of knots.Comment: 25 pages, 14 figures; references added to second versio

Quandles at Finite Temperatures I

In CJKLS quandle cohomology is used to produce invariants for particular embeddings of codimension two; 2-cocycles give to invariants for (classical) knots and 3-cocycles give rise to invariants for knotted surfaces. This is done by way of a notion of coloring of a diagram. Also, these invariants have the form of state-sums (or partition functions) used in Statistical Mechanics. By a careful analysis of these colorings of diagrams we are able to come up with new invariants which correspond to calculation of the partition function at finite temperatures.Comment: 24 pages, 4 figures, Late

The minimum number of Fox colors modulo 13 is 5

In this article we show that if a knot diagram admits a non-trivial coloring modulo 13 then there is an equivalent diagram which can be colored with 5 colors. Leaning on known results, this implies that the minimum number of colors modulo 13 is 5.Comment: 35 pages, 37 figure

On the orbits associated with the Collatz conjecture

This article is based upon previous work by Sousa Ramos and his collaborators. They first prove that the existence of only one orbit associated with the Collatz conjecture is equivalent to the determinant of each matrix of a certain sequence of matrices to have the same value. These matrices are called Collatz matrices. The second step in their work would be to calculate this determinant for each of the Collatz matrices. Having calculated this determinant for the first few terms of the sequence of matrices, their plan was to prove the determinant of the current term equals the determinant of the previous one. Unfortunately, they could not prove it for the cases where the dimensions of the matrices are 26+54l or 44+54l, where l is a positive integer. In the current article we improve on these results.Comment: 10 pages, typos correcte

Gelfand-Shilov Regularity of SG Boundary Value Problems

We show that the solutions of SG elliptic boundary value problems defined on the complement of compact sets or on the half-space have some regularity in Gelfand-Shilov spaces. The results are obtained using classical results about Gevrey regularity of elliptic boundary value problems and Calder\'on projectors techniques adapted to the SG case. Recent developments about Gelfand-Shilov regularity of SG pseudo-differential operators on $\mathbb{R}^{n}$ appear in an essential way

New Born-Oppenheimer molecular dynamics based on the extended Hueckel method: first results and future developments

Computational chemistry at the atomic level has largely branched into two major fields, one based on quantum mechanics and the other on molecular mechanics using classical force fields. Because of high computational costs, quantum mechanical methods have been typically relegated to the study of small systems. Classical force field methods can describe systems with millions of atoms, but suffer from well known problems. For example, these methods have problems describing the rich coordination chemistry of transition metals or physical phenomena such as charge transfer. The requirement of specific parametrization also limits their applicability. There is clearly a need to develop new computational methods based on quantum mechanics to study large and heterogeneous systems. Quantum based methods are typically limited by the calculation of two-electron integrals and diagonalization of large matrices. Our initial work focused on the development of fast techniques for the calculation of two-electron integrals. In this publication the diagonalization problem is addressed and results from molecular dynamics simulations of alanine decamer in gas-phase using a new fast pseudo-diagonalization method are presented. The Hamiltonian is based on the standard Extended Hueckel approach, supplemented with a term to correct electrostatic interactions. Besides presenting results from the new algorithm, this publication also lays the requirements for a new quantum mechanical method and introduces the extended Hueckel method as a viable base to be developed in the future