Aurifeuillian factorization

The Cunningham project seeks to factor numbers of the form bn±1 with b = 2, 3, . . . small. One of the most useful techniques is Aurifeuillian Factorization whereby such a number is partially factored by replacing bn by a polynomial in such a way that polynomial factorization is possible. For example, by substituting y = 2k into the polynomial factorization (2y2)2+1 = (2y2−2y+1)(2y2+2y+1) we can partially factor 24k+2+1. In 1962 Schinzel gave a list of such identities that have proved useful in the Cunningham project; we believe that Schinzel identified all numbers that can be factored by such identities and we prove this if one accepts our definition of what “such an identity” is. We then develop our theme to similarly factor f(bn) for any given polynomial f, using deep results of Faltings from algebraic geometry and Fried from the classification of finite simple groups.

Integer Points Close to a Smooth Curve

∗ This research is partially supported by the Bulgarian National Science Fund under contract MM-403/9

Frontal Solutions: an Information Technology Transfer to Abstract Mathematics

The paper introduces a method for dependencies discovery during human-machine interaction. It is based on an analysis of numerical data sets in knowledge-poor environments. The driven procedures are independent and they interact on a competitive principle. The research focuses on seven of them. The application is in Number Theory.

Powers and Logarithms

There are applied power mappings in algebras with logarithms induced by a given linear operator D in order to study particular properties of powers of logarithms. Main results of this paper will be concerned with the case when an algebra under consideration is commutative and has a unit and the operator D satisfies the Leibniz condition, i.e. D(xy) = xDy + yDx for x, y ∈ dom D. Note that in the Number Theory there are well-known several formulae expressed by means of some combinations of powers of logarithmic and antilogarithmic mappings or powers of logarithms and antilogarithms (cf. for instance, the survey of Schinzel S[1].

Fraction Space Revisited

Бойко Бл. Банчев - Знае се, че рационалните числа образуват интересни и богати на изчислителни възможности структури като редици на Фарей (Феъри) и безкрайни дървета. Малко внимание се обръща на по-общо, систематично излагане на основните свойства на дробите като множество. Понятия биват въвеждани без обосноваване, някои доказателства са ненужно изкуствени, а почти винаги и едните, и другите като че биват отнесени към една или друга особена структура, вместо към множеството на дробите изобщо. Изненадващо е, че някои същностни твърдения изглежда дори не са формулирани в литературата по теория на числата. Тази статия има за цел да подобри състоянието на нещата в това отношение, като предлага общо, подходящо подредено изложение на понятия и свързани с тях твърдения. Като допълнение са представени бележки върху пораждането на множеството от всички дроби – откритие значително по-старо, отколкото е прието да се смята.

A Fast Algorithm for Weber Problem on a Grid

AMS subject classification: 90B80.

The NumbersWithNames Program

We present the NumbersWithNames program which performs data-mining on the Encyclopedia of Integer Sequences to find interesting conjectures in number theory. The program forms conjectures by finding empirical relationships between a sequence chosen by the user and those in the Encyclopedia. Furthermore, it transforms the chosen sequence into another set of sequences about which conjectures can also be formed. Finally, the program prunes and sorts the conjectures so that themost plausible ones are presented first. We describe here the many improvements to the previous Prolog implementation which have enabled us to provide NumbersWithNames as an online program. We also present some new results from using NumbersWithNames, including details of an automated proof plan of a conjecture NumbersWithNames helped to discover.

Mathematics and fiber arts

Mathematics can be found all over the world, even in what could be considered an unrelated area, like fiber arts. In knitting, crochet, and counted-thread embroidery, we can find concepts of algebra, graph theory, number theory, geometry of transformations, and symmetry, as well as computer science. For example, many fiber art pieces embody notions related with groups of symmetry. In this work, we focus on two areas of Mathematics associated with knitting, crochet, and cross-stitch works – number theory and geometry of transformations.

Theory of modulational instability in Bragg gratings with quadratic nonlinearity

Modulational instability in optical Bragg gratings with a quadratic nonlinearity is studied. The electric field in such structures consists of forward and backward propagating components at the fundamental frequency and its second harmonic. Analytic continuous wave (CW) solutions are obtained, and the intricate complexity of their stability, due to the large number of equations and number of free parameters, is revealed. The stability boundaries are rich in structures and often cannot be described by a simple relationship. In most cases, the CW solutions are unstable. However, stable regions are found in the nonlinear Schrodinger equation limit, and also when the grating strength for the second harmonic is stronger than that of the first harmonic. Stable CW solutions usually require a low intensity. The analysis is confirmed by directly simulating the governing equations. The stable regions found have possible applications in second-harmonic generation and dark solitons, while the unstable regions maybe useful in the generation of ultrafast pulse trains at relatively low intensities. [S1063-651X(99)03005-6].

On the depth of blowup algebras of ideals with analytic deviation one

Let I be an ideal in a local Cohen-Macaulay ring (A, m). Assume I to be generically a complete intersection of positive height. We compute the depth of the Rees algebra and the form ring of I when the analytic deviation of I equals one and its reduction number is also at most one. The formu- las we obtain coincide with the already known formulas for almost complete intersection ideals.

A simple analysis of thermodynamic properties for classical plasmas: I. theory

By eliminating the short range negative divergence of the Debye–Hückel pair distribution function, but retaining the exponential charge screening known to operate at large interparticle separation, the thermodynamic properties of one-component plasmas of point ions or charged hard spheres can be well represented even in the strong coupling regime. Predicted electrostatic free energies agree within 5% of simulation data for typical Coulomb interactions up to a factor of 10 times the average kinetic energy. Here, this idea is extended to the general case of a uniform ionic mixture, comprising an arbitrary number of components, embedded in a rigid neutralizing background. The new theory is implemented in two ways: (i) by an unambiguous iterative algorithm that requires numerical methods and breaks the symmetry of cross correlation functions; and (ii) by invoking generalized matrix inverses that maintain symmetry and yield completely analytic solutions, but which are not uniquely determined. The extreme computational simplicity of the theory is attractive when considering applications to complex inhomogeneous fluids of charged particles.

An existence theorem for an analytic first order PDE in the framework of Colombeau's theory

We study the existence of a holomorphic generalized solution u of the PDE[GRAPHICS]where f is a given holomorphic generalized function and (alpha (1),...alpha (m)) is an element of C-m\{0}.

New developments in state-specific multireference coupled-cluster theory

Coupled-cluster theory in its single-reference formulation represents one of the most successful approaches in quantum chemistry for the description of atoms and molecules. To extend the applicability of single-reference coupled-cluster theory to systems with degenerate or near-degenerate electronic configurations, multireference coupled-cluster methods have been suggested. One of the most promising formulations of multireference coupled cluster theory is the state-specific variant suggested by Mukherjee and co-workers (Mk-MRCC). Unlike other multireference coupled-cluster approaches, Mk-MRCC is a size-extensive theory and results obtained so far indicate that it has the potential to develop to a standard tool for high-accuracy quantum-chemical treatments. This work deals with developments to overcome the limitations in the applicability of the Mk-MRCC method. Therefore, an efficient Mk-MRCC algorithm has been implemented in the CFOUR program package to perform energy calculations within the singles and doubles (Mk-MRCCSD) and singles, doubles, and triples (Mk-MRCCSDT) approximations. This implementation exploits the special structure of the Mk-MRCC working equations that allows to adapt existing efficient single-reference coupled-cluster codes. The algorithm has the correct computational scaling of d*N^6 for Mk-MRCCSD and d*N^8 for Mk-MRCCSDT, where N denotes the system size and d the number of reference determinants. For the determination of molecular properties as the equilibrium geometry, the theory of analytic first derivatives of the energy for the Mk-MRCC method has been developed using a Lagrange formalism. The Mk-MRCC gradients within the CCSD and CCSDT approximation have been implemented and their applicability has been demonstrated for various compounds such as 2,6-pyridyne, the 2,6-pyridyne cation, m-benzyne, ozone and cyclobutadiene. The development of analytic gradients for Mk-MRCC offers the possibility of routinely locating minima and transition states on the potential energy surface. It can be considered as a key step towards routine investigation of multireference systems and calculation of their properties. As the full inclusion of triple excitations in Mk-MRCC energy calculations is computational demanding, a parallel implementation is presented in order to circumvent limitations due to the required execution time. The proposed scheme is based on the adaption of a highly efficient serial Mk-MRCCSDT code by parallelizing the time-determining steps. A first application to 2,6-pyridyne is presented to demonstrate the efficiency of the current implementation.

Higher-order molecular properties and excitation energies in single-reference and multireference coupled-cluster theory

Coupled-cluster (CC) theory is one of the most successful approaches in high-accuracy quantum chemistry. The present thesis makes a number of contributions to the determination of molecular properties and excitation energies within the CC framework. The multireference CC (MRCC) method proposed by Mukherjee and coworkers (Mk-MRCC) has been benchmarked within the singles and doubles approximation (Mk-MRCCSD) for molecular equilibrium structures. It is demonstrated that Mk-MRCCSD yields reliable results for multireference cases where single-reference CC methods fail. At the same time, the present work also illustrates that Mk-MRCC still suffers from a number of theoretical problems and sometimes gives rise to results of unsatisfactory accuracy. To determine polarizability tensors and excitation spectra in the MRCC framework, the Mk-MRCC linear-response function has been derived together with the corresponding linear-response equations. Pilot applications show that Mk-MRCC linear-response theory suffers from a severe problem when applied to the calculation of dynamic properties and excitation energies: The Mk-MRCC sufficiency conditions give rise to a redundancy in the Mk-MRCC Jacobian matrix, which entails an artificial splitting of certain excited states. This finding has established a new paradigm in MRCC theory, namely that a convincing method should not only yield accurate energies, but ought to allow for the reliable calculation of dynamic properties as well. In the context of single-reference CC theory, an analytic expression for the dipole Hessian matrix, a third-order quantity relevant to infrared spectroscopy, has been derived and implemented within the CC singles and doubles approximation. The advantages of analytic derivatives over numerical differentiation schemes are demonstrated in some pilot applications.